Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 64026
Multiplying Polynomials - 2
Students are asked to multiply polynomials and explain if the product of two polynomials always results in a polynomial.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, polynomials, multiplication, product, closure, like terms
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_MultiplyingPolynomials-2_Worksheet.pdf
MFAS_MultiplyingPolynomials-2_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 problems on the Multiplying Polynomials - 2 worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is not able to correctly multiply the polynomials and is unable to determine or explain that the polynomials are closed for multiplication.
Examples of Student Work at this Level
The student:
Does not correctly distribute. Answers may include:
page 1 of 3 Multiplies the expressions partially or fully but then combines terms that are not like or adds exponents. Answers may include:
Questions Eliciting Thinking
What does it mean to distribute?
How did you multiply the two binomials?
How do you square a binomial? Did you rewrite it as a multiplication first?
There is an error in your multiplication; can you find it?
Instructional Implications
Review the application of the Distributive Property with the student. Model the use of the Distributive Property when multiplying two binomials. Explain that the Distributive
Property is used twice, so both terms of the second binomial are multiplied by each of the terms of the first binomial. When the student is proficient using this approach,
guide the student to discover and use more efficient methods such as “FOIL.” Provide problems for the student to complete.
When squaring a binomial, guide the student to first rewrite the binomial as the product of two factors,
, and then multiply using the same
approach as when multiplying two binomials. Eventually, guide the student to observe the relationship between the terms of the binomial base and the terms of its
expansion [e.g.,
]. Assist the student in using this relationship to square binomials. Provide additional problems for the student to complete.
Once the student is proficient multiplying polynomials, review the concept of closure of the integers under addition, subtraction, and multiplication. Then introduce the
concept of closure of the polynomials under addition, subtraction, and multiplication.
Making Progress
Misconception/Error
The student correctly multiplies the polynomials but is unable to adequately explain that the polynomials are closed for multiplication.
Examples of Student Work at this Level
The student answers one question positively and one negatively or both negatively (questions 3 and 4). The student does not demonstrate an understanding of a
polynomial.
The student answers both questions (3 and 4) positively, but cannot explain why the answers are polynomials.
Questions Eliciting Thinking
What is a polynomial?
What must be true of the terms of a polynomial? What do they look like?
Can you give me an example of an expression that is not a polynomial?
Is the expression 0 a polynomial?
Instructional Implications
Review the definition of a polynomial as the sum or difference of terms of the form
where a is a real number and n is a whole number. The prefix poly can be misleading
since a polynomial may have only one term, which could be a constant including zero. Provide additional examples and non-examples of polynomials (e.g., 0,
9,
,
, 3, y -
). Ask the student to identify the examples of polynomials and explain why each non-example fails the definition.
Model explaining that for any two terms
and
(where a and b are real numbers and m and n are whole numbers),
. Since
the real numbers are closed for multiplication, (a · b) is a real number. Since the whole numbers are closed for addition, (m + n) is a whole number. So, multiplying terms of
polynomials results in terms that fit the definition of a polynomial [i.e., (a · b) is a real number and (m + n) is a whole number].
Consider implementing MFAS tasks Subtracting Polynomials (A-APR.1.1), Multiplying Polynomials - 1 (A-APR.1.1), and Adding Polynomials (A-APR.1.1) to further assess the
student's understanding of the closure of polynomials under the operations of addition, subtraction, and multiplication.
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 responds to the four questions as follows:
1.
2.
3. The student explains that
number. Likewise,
is a polynomial since it consists of the sum of two terms each of the form
where a is a real number and n is a whole
is a polynomial since the coefficients are real numbers for each term and all of the exponents are whole numbers.
4. The student further explains that the product of two polynomials will always be a polynomial because the result of multiplying terms of the form
of the form
. The student may show that for any two terms
and
results in more terms
(where a and b are real numbers and m and n are whole numbers),
. Since the real numbers are closed for multiplication, (a · b) is a real number. Therefore, multiplying terms of polynomials results in
terms that fit the definition of a polynomial.
page 2 of 3 Questions Eliciting Thinking
When you add or subtract two polynomials, will the result always be a polynomial?
When you divide two polynomials, will the result always be a polynomial?
Instructional Implications
Challenge the student to find an example of an operation with two polynomials that will result in an expression that is not a polynomial.
Consider implementing MFAS tasks Subtracting Polynomials (A-APR.1.1), Multiplying Polynomials - 1 (A-APR.1.1), and Adding Polynomials (A-APR.1.1) to further assess the
student"s understanding of the closure of polynomials under the operations of addition, subtraction, and multiplication.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Multiplying Polynomials - 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-APR.1.1:
Description
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations
of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Remarks/Examples:
Algebra 1 - Fluency Recommendations
Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in algebra, as
well as in their symbolic work with functions. Manipulation can be more mindful when it is fluent.
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