Instructional Leadership Initiative: Supporting Standards-based Practice
Unit:
Algebra 1: Polynomials
School:
Pasadena High School
Strengths:
This unit contains a well-formatted assessment that is aligned with the content
standard and provides the type of evidence required by that standard.
Concerns:
It appears that this unit underestimates the number of separate prerequisite skills
that students must possess in order to meet the requirements of this content
standard. it also appears that some of the prerequisite skills have been included in
the unpacking of the standard.
There is not great alignment of opportunities to learn/practice with the demands
made of students through the assessment. One example of this is that students are
required to identify errors in reasoning for three assessment items but there is no
mention of students having practice in doing so prior to the assessment.
The rubric would enable students to “meet the standard” even if they had
incomplete or incorrect solutions to problems. It is unclear what “infrequent minor
errors” means. It is also unclear what “insignificant errors” means. The subjectivity
of these descriptions could allow some students to be judged as “meeting the
standard” and others as “not meeting the standard” even if they had similar papers.
Pasadena – Algebra – Polynomials
Revised 08-15-2002
Page 1
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
I.
BACKGROUND
Unit Title:
Grade Level: 9-12
Polynomials
Anthony Carruthers, Reyna Guzman, Harriet
Unit
Hammond, Kaia Mashariki, Osvaldo Mejia,
Designers: Frank Scoonover
Discipline/Course Title: Mathematics, Algebra I
Timeframe:
3-5 Days – 1 hour Sessions
Teacher to Teacher Notes:
This unit encompasses monomials and polynomials by successive
steps. In the course of working with polynomials, students will
learn adding, subtracting, multiplying, and dividing polynomials.
This particular unit will cover part of Algebra 1 Standard 10,
“Students will add, subtract and multiply monomials and
polynomials. Students solve multi-step problems, including word
problems, by using these techniques.” New vocabulary words will
be learned and used. Real-life geometric problems involving area
and perimeter will be used. Students will use algebra tiles to
conceptualize mathematical ideas.
This unit is composed of 3-5 multi-day core lessons. The amount
of time needed for each lesson will depend in part on prior
knowledge of l.c.m., practical problem-solving, solving equations,
area and perimeter. Each of the lessons results in student work that
can be used as diagnostics to plan future instruction.
The time required for completing
this standard seems to be
unclear. Is it “3-5 days” or “3-5
multi-day core lessons?” most
textbooks devote several
chapters to addition, subtraction,
and multiplication of monomials
and polynomials as this
comprises a good portion of an
Algebra course.
While working in different sections, students will become
proficient with the use of algebra tiles to add, subtract and multiply
polynomials. Using the manipulatives and other methodologies,
students will practice solving problems in engineering,
communication and economics. Properties of polynomials make it
possible to find most efficient use of time and materials.
Students will be required to complete one assessment. Part I
consists of skills-based examples, while Part II consists of multistep word problems. This assessment will consist of point scores
and a rubric for the open-ended questions.
Printed Materials
Needed:
Algebra Tiles, Polynomials
Assessment, Worksheets #1-6.
Resources (non-print):
Internet Resources:
None
None
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Pasadena – Algebra – Polynomials
Revised 08-15-2002
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
II. CONTENT STANDARDS ADDRESSED
The required content knowledge
State of California/Pasadena Unified School
State/District: District
Title:
Polynomials
Algebra I Standard 10: Students add, subtract, and multiply
monomials and polynomials. Students solve multi-step problems,
including word problems, by using these techniques.
Unpack the Standard
The knowledge and skills that students need to know and be
able to do to meet the standard
1. Given a polynomial or monomial, find each sum or difference
2. Given a polynomial or monomial, multiply
3. Write a polynomial expression for the perimeter of a given
geometric figure
4. Find the expression for the area of a shaded region of a
geometric figure
5. State the degree of polynomials
6. Write polynomial expressions in descending order
7. State the number of terms there are in a given polynomial
8. Read and interpret word problems
Several skills listed here are prerequisites. Items 3 and 4 are not
inherent to the standard but
applications of the standard; also,
word problems can and should be
more than application of area and
perimeter formulas. In general, a
rephrasing of the unpacking of the
standard would be appropriate:
1. Write polynomials in standard
form (with exponents of terms
in descending order).
2. Given a set of two or more
monomials and/or
polynomials, find the sum by
combining like terms.
3. Given two monomials, two
polynomials, or a polynomial
and monomial, find the
differences.
4. Multiply a monomial times
another monomial or
polynomial.
5. Multiply a polynomial by
another polynomial.
6. Solve word problems requiring
addition, subtraction, or
multiplication of monomials
and/or polynomials.
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Pasadena – Algebra – Polynomials
Revised 08-15-2002
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
Enabling Prerequisite Skills:
•
•
•
•
•
Understands least common multiple
Solves practical problem-solving
Solves linear equations
Understands area and perimeter
Understands exponents
Teacher to Teacher Notes:
As you unpack the standard, look through different resources to
find additional multi-step problems. In writing our assessment, our
group of teachers found that it’s not easy to find problems that
assess student learning. As you talk about student assessment,
unpacking the standard becomes a very productive exercise.
There are additional prerequisites
for this standard. Students will
need to bring the following skills
with them:
• Mastery of operations (+, -, x, ÷)
on integers
• Applies distributive property
• Identifies least common multiple
of two or more terms
• Uses operations on positive and
negative integers correctly
• Uses laws of exponents to
multiply terms with like variables
• Identifies degree of polynomials
• Simplifies expressions correctly
• Solves multi-step problems,
including word problems,
involving linear equations
• Uses algebraic terminology
(e.g., variable, term, coefficient,
expression, constant, exponent,
power) correctly
• Represents quantitative
relationships algebraically
• Able to calculate area and
perimeter of polygons
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Pasadena – Algebra – Polynomials
Revised 08-15-2002
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
III. THE ASSESSMENT
What students will need to do to provide evidence
that they have met the standard.
Type(s) of Evidence Required to Assess the
Standard(s):
•
•
•
•
•
•
Ability to add, subtract, and multiply monomials and
polynomials
Ability to read and interpret word problem
Ability to recognize the number of terms in an expression
Ability to identify the degree of a monomial or polynomial
Ability to use the distributive property to simplify expressions
Ability to write monomial and polynomial expressions for
perimeters and areas of basic geometric shapes
The term “ability” seems
inappropriate here. This heading
asks for a description of evidence
– usually a work-product – not an
assessment of whether a student
has capability.
Assessment Method(s):
•
•
Open-response assessment
Multiple choice questions
Teacher to Teacher Notes:
Students will be required to complete the assessment in 45
minutes. Students are not allowed to use notes or calculators.
Assessment Prompt(s):
•
•
•
•
This is a rather lengthy test to be
finished in 45 minutes. There is no
mention here of any
accommodations for students with
special needs (e.g., LD, ELL) or
students who may just work
slowly
Show all work neatly on space provided for Part I.
You may use more paper to fully answer each question if
needed.
Students must show all work neatly on a separate sheet of
paper for Part II.
This is an independent assessment. Work alone.
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Pasadena – Algebra – Polynomials
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Western Assessment Collaborative at
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Name ______________________
Period ________
Date _____
POLYNOMIALS TEST – REVISED
(addition, subtraction and multiplication)
PART I
3 x − 6 x + 2 x +1
2
4
2. Write in descending order, − x + 2 x + 5 − 8 x
2
3. How many terms are there in the expression 3 x + 4 − 5 y − 7 x
5
1. State the degree of the polynomial,
2
4. Which of the following is not a polynomial ?
a. x 2 + y 2
1
3
c.
x2 −
e.
1
+y
x
1. ______________
2. ______________
3. ______________
b.
x3
d.
ax 2 + bx + c
4. ______________
Perform the indicated operation. Write your answer in standard form.
5.
7.
( 6 x + 3) + ( −5 − 4 x )
( 4a − 2 + 12b ) − ( 7a + 14 − 3b )
9.
( x − 1)( 3x + 4 )
6.
( 3x
8.
( 3a 2 + a ) − ( −a 2 − 8a + 7 )
10.
2
+ 9 x − 11) + ( − x 2 − 20 )
(b − 2)
2
Items 1 through 4 appear to be
appropriate for scaffolding student
thinking as they focus on
prerequisite skills.
For diagnostic purposes, it might
be helpful to actually have an item
involving multiplication of two or
more monomials as well as
multiplication of a polynomial by a
monomial. If these simpler
problems are not completed
correctly, the teacher would have
an easier time of remediating the
student than if he/she tries to
figure out why any of items 5-14
are incorrect.
Items 5-14 have adequate space
for students to do the work
involved. There is no request for
all steps to be shown, so it would
be possible for some students to
perform the operations mentally
or on scratch paper and just show
the result. If work is to be shown,
the directions should be explicit.
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Pasadena – Algebra – Polynomials
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Western Assessment Collaborative at
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Perform the indicated operation. Write your answer in standard form.
11.
( 3a + 2 ) ( 2a 2 − a + 3)
12.
( x + 2 y )( x − 2 y )
13.
8 + 3( x − 2)
14.
5 ( x + 3) − 3 ( 4 − 2 x )
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Pasadena – Algebra – Polynomials
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Instructional Leadership Initiative: Supporting Standards-based Practice
PART II
Complete all questions in the space provided. Show all steps for full credit.
15. Your neighbor asks you to cut her grass. Her yard is (6x + 3) feet wide and
(10 x + 3) feet long. A section of the yard is fenced in for her dog. The dog
cage is (x – 2) feet wide and (x + 2) feet long.
Problems 15 and 17 appear to be
very similar.
Are there applications involving
adding, subtracting, and
multiplying monomials and
polynomials other than area and
perimeter?
Dog
Cage
a.
Write a polynomial expression that represents the total area of the yard.
Give your answer as a polynomial expression.
b.
Write a polynomial expression that represents the area of the dog cage.
Write your answer as a polynomial expression.
c.
Write a polynomial expression that represents the area of the yard that you
will mow.
Give your answer as a polynomial expression.
16. The perimeter of a triangle is 231 inches. The lengths of the sides are three
consecutive odd integers. Find the length of each side of the triangle.
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Pasadena – Algebra – Polynomials
Revised 08-15-2002
Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
17. Uncle Robert’s swimming pool is (3d – 4) feet long d feet wide
He wants to build a fence around the rectangular pool deck.
Uncle Robert wants the fence dimensions to be (21d + 8) feet
long and 2d feet wide. What is the area of the deck around the
pool ?
21d + 8
d
2d
3d - 4
Look over problems 18 – 20.
(a.) Circle all errors.
(b.) Simplify the expressions correctly in the space provided. Show
all work.
18 (a.)
(3x + 8) – (5x – 2)
18 (b.)
= 3x + 8 – 5x – 2
= 2x + 6
Identifying errors is an important
skill as well as a useful way of
assessing mastery. If students
have had practice with similar
items prior to the assessment,
these would appear to be very
good items.
19 (a.)
(5a + 4b) + (3a + 2c) = 5a + 3a + 4b + 2c
= 8a2 + 6bc
19 (b.)
20 (a.)
(7n + 3k)(2n + k)
20 (b)
= 7n(2n) + 3k(k)
= 14n + 4k2
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Pasadena – Algebra – Polynomials
Revised 08-15-2002
Western Assessment Collaborative at
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This is an incorrect answer. The
correct answer is 2x2 + 9x – 31
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Pasadena – Algebra – Polynomials
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Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
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Pasadena – Algebra – Polynomials
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Western Assessment Collaborative at
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Pasadena – Algebra – Polynomials
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Western Assessment Collaborative at
Instructional Leadership Initiative: Supporting Standards-based Practice
The term “8a2” is also
incorrect.
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Pasadena – Algebra – Polynomials
Revised 08-15-2002
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Instructional Leadership Initiative: Supporting Standards-based Practice
IV. CRITERIA FOR SUCCESS
What will be expected of the students on the
assessment
Characteristics of a High Quality Response to the
Assessment:
The student produces a response that:
•
•
•
•
•
Correctly combines like terms to simplify the expression for
perimeter or area
Correctly adds, subtracts, and multiplies polynomials
Correctly simplifies expressions by grouping like terms
Carries out appropriate operations on polynomials
Recognizes, identifies and circles errors in problems 18 - 20
and simplifies the expressions correctly
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Pasadena – Algebra – Polynomials
Revised 08-15-2002
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Instructional Leadership Initiative: Supporting Standards-based Practice
V.
OPPORTUNITIES TO LEARN AND PERFORM
Instructional plan to assure that every student has
adequate opportunities to learn and practice what is
expected.
What vocabulary terms are to be
incorporated within this unit?
Opportunities to Learn:
•
•
•
•
•
•
Lesson on definitions of vocabulary terms
Lesson on stating the number of terms in a given polynomial
Lesson on writing polynomials in descending order
Lesson on addition, subtraction of polynomials
Lesson on multiplication of polynomials
Lesson on interpreting, setting up, and solving word problems
where it is necessary to find the perimeter or area of the given
geometric figure
Will there be any opportunities to
learn how to approach word
problems that are not area or
perimeter problems?
Since identification of errors is
assessed, it would seem
appropriate to provide students
with practice in doing so prior to
the written assessment.
Opportunities to Perform:
•
•
•
•
•
•
Practice definitions of vocabulary terms
Practice stating the number of terms in a given polynomial
Practice writing polynomials in descending order
Practice addition, subtraction of polynomials
Practice multiplication of polynomials
Practice interpreting, setting up, and solving word problems
using the perimeter or area of given geometric figures.
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Pasadena – Algebra – Polynomials
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Instructional Leadership Initiative: Supporting Standards-based Practice
VI. THE PERFORMANCE STANDARD
Rubric or other form of scoring guide
To Standard
LEVEL 5
• Meets all standards.
• Student correctly adds, subtracts and multiplies monomials and
polynomials, making infrequent minor errors unrelated to the
standard.
• Correctly simplifies by combining like terms.
• Demonstrates competence in interpreting and solving multistep problems, including word problems.
• Executes step by step calculations, clearly stating answers to
the problem and checking solutions
• Student implicitly or explicitly defines the variables involved
in multi-step problems.
LEVEL 4
• Meets all standards
• Student correctly adds, subtracts and multiplies monomials and
polynomials, making infrequent minor errors unrelated to the
standard.
• Correctly simplifies by combining like terms.
• Correctly solves at least one multi-step problem with
insignificant errors; if any.
Not to Standard
LEVEL 3
• Student correctly performs two of the three basic operations
with monomials and polynomials (i.e. addition, subtraction and
multiplication), making infrequent minor errors unrelated to the
standard.
• Student correctly simplifies answers by combining like terms.
• Student demonstrates some understanding of solving multi-step
problems allowing for errors such as:
a. Setting up algebraic expressions incorrectly
b. Failing to solve algebraic equations correctly
c. Giving implausible answers are given.
d. Giving illogical answers
e. Failing to complete operations or simplify expressions
correctly
The first bullet for levels 5 and 4
requires that the student “Meets
all standards.” The purpose of
the rubric is to provide a way for
measuring how well the student
meets the standard. What does
“Meets all standards” mean?
What is the definition of a “minor
error”? How frequent can errors
be made to be considered
“infrequent”?
Had students been directed to
show all steps or check their
solutions on the assessment,
then “executing step by step
calculations, clearly stating
answers to the problem and
checking solutions” would be
appropriate for inclusion in the
rubric.
How does one really differentiate
between implausible and illogical
answers?
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Pasadena – Algebra – Polynomials
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LEVEL 2
• Correctly performs one of the three basic operation with
monomials and polynomials (i.e. addition, subtraction and
multiplication) making infrequent minor errors unrelated to the
standard.
• Attempts to combine like terms.
LEVEL 1
• Fails to meet levels 2, 3, 4 or 5
• Demonstrates limited, but insufficient
competence/understanding, if any, of the three basic
operations.
• Demonstrates complete competence for none of the three basic
operations.
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Pasadena – Algebra – Polynomials
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Instructional Leadership Initiative: Supporting Standards-based Practice
VII. SAMPLES OF STUDENT WORK WITH
COMMENTARY
REVISED
Commentary – Overview:
We have revised the unit since receiving the group’s feedback at
the June meeting: “Instructional Leadership Initiative – Supporting
Standards–based Practice”. The group gave us many suggestions
and ideas, which we used to improve the unit.
We have now divided the standard into two parts; this unit covers
the first part: “Identifying polynomials” and “Adding, Subtracting
and Multiplying polynomials”. The second part of the standard,
“Dividing polynomials”, is to be treated as a separate unit.
Secondly, we have replaced some problems on the assessment with
more appropriate ones; other problems have been omitted
altogether because they did not pertain to the part of the standard
covered by the revised unit.
In revising our unit, it became apparent that the student work we
had used previously did not coincide with our revised unit. Since
several teachers were teaching this unit as part of their Summer
Curricula, we were able to collect and analyze student work
specifically based on the revised unit. The new student work
samples have been included.
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Pasadena – Algebra – Polynomials
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MEETS THE STANDARD:
Example: Level 5
Commentary
#5, #6 Student
correctly adds
polynomials
#7, #8 Student
correctly subtracts
polynomials
#9, #10 Student
correctly multiplies
binomials and
correctly simplifies
by combining like
terms
The student does not correctly
note that a quantity is being
subtracted through use of
parenthesis and most likely will
have problems in the future.
(e.g., – – a2 – 8a + 7)
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Pasadena – Algebra – Polynomials
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Example: Level 5 cont.
Commentary
Correctly multiplies
polynomials
Correctly simplifies
by combining like
terms
Page 20
Pasadena – Algebra – Polynomials
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Example: Level 5 cont.
Commentary
Demonstrates
competence in
interpreting and
solving multi-step
problems, including
word problems
Executes step by step
calculations, clearly
stating answers to the
problem
Student implicitly or
explicitly defines the
variables involved in
multi-step problems
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Pasadena – Algebra – Polynomials
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Example: Level 5 cont.
Commentary
#19(b.) and #20(b.)
Student correctly
adds polynomials,
making infrequent or
minor errors
unrelated to the
standard
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Pasadena – Algebra – Polynomials
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MEETS THE STANDARD:
Example: Level 4
Commentary
Student correctly
adds polynomials
Student correctly
subtracts polynomials
Student correctly
multiplies binomials
and correctly
simplifies by
combining like terms
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Pasadena – Algebra – Polynomials
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Example: Level 4 cont.
Commentary
Student did not
attempt this page but
showed enough
evidence to merit
level 4
Student only performed two
multiplication problems correctly.
At this point, the student has not
demonstrated ability to multiply a
binomial times a trinomial, or to
multiply difference of squares,
nor perform multi-step problem
that includes addition as well as
multiplication. It is debatable as
to whether there is enough
evidence to state that the student
has really met the standard.
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Pasadena – Algebra – Polynomials
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Example: Level 4 cont.
Commentary
Correctly solves at
least one multi-step
problem with
insignificant errors, if
any
The directions say that the
student must “show all steps for
full credit.” How much credit was
possible for item 16 since the
solution of 75, 77, & 79 is correct
and could have been intuitively
derived?
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Pasadena – Algebra – Polynomials
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Example: Level 4 cont.
Commentary
Student shows more
evidence of correctly
adding, subtracting,
and multiplying
polynomials
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DOES NOT YET MEET THE STANDARD:
Example: Level 3
Commentary
#5, #6 Student
correctly adds
polynomials
#7, #8 Student
correctly subtracts
polynomials
#9, #10 Student fails
to multiply correctly
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Example: Level 3 cont.
Commentary
It is interesting to note that this
student was unable to complete 3
of the 4 multiplication items, yet
was able to correctly find the
product in item 12. is this related
to a possible attendance pattern?
Student correctly
simplifies answers by
combining like terms
#13, #14
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Example: Level 3 cont.
Commentary
Demonstrates some
understanding of
solving multi-step
problems #15 a
The student understood that the
length and width needed to be
multiplied to find area but was
unable to combine like terms.
Again, the student only
successfully finds the product of
the difference of squares.
Implausible answers
in #15 c
Interesting to see that this
student was able to set up the
solution in item 16.
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Example: Level 3 cont.
Commentary
Student sets up
algebraic expressions
incorrectly
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DOES NOT YET MEET THE STANDARD:
Example: Level 2
Commentary
#5, #6 Student
correctly adds
polynomials making
infrequent minor
errors unrelated to
the standard (#6)
Student fails to
subtract polynomials
(#7, #8)
Student fails to
multiply polynomials
(#9, #10)
This student would benefit from
reteaching focused on the use of
parenthesis when subtraction of
quantities are involved. It appears
that subtraction of a quantity that
has more than one term is
unclear.
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Pasadena – Algebra – Polynomials
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Example: Level 2 cont.
Commentary
Student fails to
multiply polynomials
Student fails to
simplify expressions
by combining like
terms
Actually, the student correctly
combined like terms in items 12
& 14. The student incorrectly
combined unlike terms in item
13.
Page 32
Pasadena – Algebra – Polynomials
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Example: Level 2 cont.
Commentary
Student attempts
word problem but
fails to multiply
correctly and
comprehend the last
part (#15 c)
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Example: Level 2 cont.
Commentary
This student had no idea what
was being asked for – there was
no reason to attempt a solution in
18b.
Student shows more
evidence of adding
polynomials (#19 b)
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Pasadena – Algebra – Polynomials
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DOES NOT YET MEET THE STANDARD:
Example: Level 1
Commentary
Student fails to add
polynomials correctly
Student fails to
subtract polynomials
correctly
Not only does the student fail to
follow directions and perform the
indicated operations, the student
appears to have no concept of
“like terms” or when a solution is
warranted.
Student fails to
multiply polynomials
correctly
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Pasadena – Algebra – Polynomials
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Example: Level 1 cont.
Commentary
Student fails to
multiply polynomials
correctly
Student fails to
simplify expressions
by combining like
terms
Page 36
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Example: Level 1 cont.
Commentary
Student could not
interpret or solve the
word problems
correctly
It is difficult to see if the student
understands what is meant by
“consecutive odd integers.”
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Example: Level 1 cont.
Commentary
Student could not
interpret word
problem correctly
Student failed to
correctly add,
subtract, and multiply
polynomials.
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Student Assessment
Page 39
Pasadena – Algebra – Polynomials
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Western Assessment Collaborative at
Name ___________________________________
POLYNOMIALS TEST
Period ________
Date ____________________
(addition, subtraction and multiplication)
PART I
5.
6.
7.
8.
State the degree of the polynomial, 3x − 6 x 5 + 2 x 2 + 1
Write in descending order, − x + 2 x 2 + 5 − 8 x 4
How many terms are there in the expression 3x + 4 − 5 y − 7 x 2
Which of the following is not a polynomial ?
b. x 3
a. x 2 + y 2
1
c. x 2 −
d. ax 2 + bx + c
3
1
e.
+y
x
5. ______________
6. ______________
7. ______________
8. ______________
Perform the indicated operation. Write your answer in standard form.
5. ( 6 x + 3) + ( −5 − 4 x )
6. ( 3x 2 + 9 x − 11) + ( − x 2 − 20 )
7.
( 4a − 2 + 12b ) − ( 7a + 14 − 3b )
9.
( x − 1) ( 3x + 4 )
8.
10.
( 3a
2
+ a ) − ( −a 2 − 8a + 7 )
(b − 2)
2
Perform the indicated operation. Write your answer in standard form.
11. ( 3a + 2 ) ( 2a 2 − a + 3)
12. ( x + 2 y )( x − 2 y )
13. 8 + 3 ( x − 2 )
14. 5 ( x + 3) − 3 ( 4 − 2 x )
PART II
Complete all questions in the space provided. Show all steps for full credit.
17. Your neighbor asks you to cut her grass. Her yard is (6x + 3) feet
wide and (10 x + 3) feet long. A section of the yard is fenced in for
her dog. The dog cage is (x – 2) feet wide and (x + 2) feet long.
Dog
Cage
d. Write a polynomial expression that represents the total area of the
yard. Give your answer as a polynomial expression.
e. Write a polynomial expression that represents the area of the dog cage.
Write your answer as a polynomial expression.
f. Write a polynomial expression that represents the area of the yard that you will mow.
Give your answer as a polynomial expression.
18. The perimeter of a triangle is 231 inches. The lengths of the sides are three consecutive odd
integers. Find the length of each side of the triangle.
17. Uncle Robert’s swimming pool is (3d – 4) feet long d feet wide
He wants to build a fence around the rectangular pool deck.
Uncle Robert wants the fence dimensions to be (21d + 8) feet
long and 2d feet wide. What is the area of the deck around the
pool ?
21d + 8
d
ddd
Look over problems 18 – 20.
(c.) Circle all errors.
(d.) Simplify the expressions correctly in the space provided. Show all work.
18 (a.)
(3x + 8) – (5x – 2) = 3x + 8 – 5x – 2
18 (b.)
= 2x + 6
19 (a.) (5a + 4b) + (3a + 2c) = 5a + 3a + 4b + 2c
19 (b.)
= 8a2 + 6bc
20 (a.) (7n + 3k)(2n + k) = 7n(2n) + 3k(k)
= 14n + 4k2
20 (b)
3d - 4
dd d
2d
Instructional Leadership Initiative: Supporting Standards-based Practice
Assessment
Answer Key
Page 44
Pasadena – Algebra – Polynomials
Revised 08-15-2002
Western Assessment Collaborative at