Download Algebra: Adding, Subtracting, and Multiplying Polynomials

Instructional Leadership Initiative: Supporting Standards-based Practice
Unit:
Algebra: Adding, Subtracting, and Multiplying Polynomials
School:
Mar Vista High School
Strengths:
The annotations which accompany the anchor papers display deep analysis of
student work. The clear explanations provide a rationale for the ratings. If this
level of analysis is used as formative feedback for instructional purposes, both
students and teachers will benefit.
Several of the items on the revised assessment require that students do more than
mechanically perform an algebraic operation. Students are required to actually
think about what information they have, what needs to be done, and then how to
interpret and use their result.
Opportunities to learn and perform, as described in core lessons, would appear to
provide rich experiences for students and lead to mastery of the standard.
Concerns:
The rubric does not allow for clear ratings of student papers. Much of the
description of the characteristics is vague and subjective. It is possible for students
with only minimal grasp of the standard to be deemed “meeting the standard”.
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 1
Instructional Leadership Initiative: Supporting Standards-based Practice
I.
BACKGROUND
Unit Title:
Adding, Subtracting &
Multiplying Polynomials
Unit Designers:
From Mar Vista High: Mary Young, Glenn
Boardman, Ricardo Gomez
From Mar Vista Middle: Colleen Miller, Beth
Lovell, Bill Walsh
Discipline/Course Title:
Timeframe:
Grade Level: 8-9
Algebra 1/2
5-7 class periods
Teacher to Teacher Notes:
After completing this unit, students will be able to add or subtract
polynomial expressions. Several approaches will be presented to
provide access for different learning styles. (e.g. horizontal and vertical
for adding and subtracting, area model and FOIL for multiplying).
Students will also become proficient at finding the product of two
polynomials and squaring a binomial. Although not specifically
mentioned in either the state or district versions of the standard
addressed, students will be expected to express answers in "standard"
form, as this is the way the answers are presented on standardized tests.
This topic is part of California Algebra 1 Standard 10.0. The
Sweetwater District has divided California's Standard 10 into two parts,
and the part addressed in this unit addresses addition, subtraction and
multiplication only. Division of polynomials is covered as a separate
topic in a different district standard.
Printed Materials Needed:
Copies of worksheets and copies of
the assessments
Algebra Tiles (overhead & student
versions)
Resources (non-print):
Text: McDougal Littell’s Algebra
1 (Section 10.1 - 10.3)
Internet Resources:
(none)
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 1
Instructional Leadership Initiative: Supporting Standards-based Practice
II. CONTENT STANDARDS ADDRESSED
The required content knowledge
State/District: Sweetwater Union High School District.
Title: ADDING, SUBTRACTING & MULTIPLYING POLYNOMIALS
CA Algebra 1 Standard 10.0: Add, subtract, and multiply monomials
and polynomials. Students solve multi-step problems, including word
problems, by using these techniques.
Sweetwater District Algebra 1/2 Standard 6.1: Add, subtract, and
multiply polynomials.
Since there is no
presentation of what an
“unpacking” of the standard
would include, it is difficult
to know what meeting the
standard would entail.
Enabling Prerequisite Skills:
Students are able to:
What are the prerequisite

perform operations with rational numbers, including signed numbers
vocabulary terms? What

use the distributive property
other essential concepts

understand exponent notation, including the rules for adding and
multiplying (5x2 + 3x2 = 8x2 ; 5x 3x = 15x2)



use the area formula for rectangles (for "word" problems)
are required of students
prior to beginning the new
learnings (e.g., perimeter,
like terms, etc.)?

understand simple variable expressions i.e. recognize 5x as "5 times
x”, differentiate between “x x” and “x+x”, and be able to identify and
combine like terms
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 2
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):
(See separate student-ready end-of-unit assessment)
Assessment Method(s):
This description of the
assessment methods does
Students will be given two formative constructed-response assessments
(Quiz A & Quiz B) and the summative (final) constructed-response
assessment
Teacher to Teacher Notes:
not provide an idea of what
students would need to
show and do to prove they
meet the standard.
Be sure to write liberal corrective remarks on student papers for quizzes A
and B, and go over commonly made errors with the whole class before
giving the summative assessment. Students will probably need 20-30
minutes for the separate, summative "Polynomials Assessment" and
considerably less time for each of the two shorter quizzes. "Special"
products (i.e. the square of a binomial and products should as (a+b)(a-b))
are covered, but heavy emphasis should not be placed on memorization of
these forms, since the same techniques used for other polynomial products
will also work for these products.
The teacher-to-teacher
note provides some useful
reminders for teachers
about formative feedback.
Assessment Prompt(s):
The summative assessment is included as a separate student-ready
document. Below are the problems for the formative assessments (Quiz A
and Quiz B). No calculators or algebra tiles may be used during any of the
quizzes. ( “Student-ready” versions of these two quizzes should be typed
to include adequate space for students to complete the problems.)
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 3
Formative Quiz A (Following Lesson 2)
Find each sum and express answers in standard form.
1. (8j - 3k - 6m) + (-2j + 3m)
2.
(3d2 - 6d - 2d3 +1) + (-d3 + 2d - 5d2 - 5)
Ans: 6j -3k -3m
Ans: -3d3 - 2d2 - 4d - 4
Find the difference and express answers in standard form:
3.
(7n3 - 8) - (6n3 - n2 - 10)
Ans: n3 + n2 +2
4.
(6x3 - 6) - (x2-8x + 9)
Ans: 6x3 -x2 + 8x -15
Complete the indicated operations and express answers in simple form:
5. (4x3 -7x2 + 3x + 9) + (x3 + 8x2 + 5x -1 ) - (x4 - 4x3 + 7x)
Ans: -x4 + 9x3 + x2 + x + 8
6. For the triangle shown, write an expression for the perimeter.
2x+3r
3x-r
4x+2r
Page 4
Formative Quiz B (Following Lesson 4)
Multiply and express answers in standard form.
x (8x  16xy  7y )
3
2
2
Ans: 2x  4x y 
5
4
7 3 2
xy
4
1.
1
4
2.
(5a + 1)(2a - 3)
Ans: 10a2 - 13a - 3
3.
(m + 5) (8m + 3m2 - 2m3)
Ans: -2m4 - 7m3 + 23m2 + 40m
4. (3y - 7)2
Ans: 9y2 - 42y +49
5. Find the area of a rectangle that has a length of 5x + 1 and a width of 2x + 3.
Ans: 10x2 + 17x +3
Page 5
Polynomials Assessment (+, -,  ) Form A
Per. ___ Name _________________________ Row ___
Teacher ______________________ Date ______
Algebra 2
SUHSD Standard 6.1:
Add, subtract, and multiply polynomials.
CA Standard 10.0:
Add, subtract, multiply, and divide monomials and polynomials.
Students solve multi-step problems, including word problems, by
using these techniques.
SUHSD
Standard
6.1
Text
Sec.
For # 1-10: Simplify and write your answers in
standard form.
10.1
1. (5n  r)  (7n  3r)
6.1
2.

1
2
x 2  53 x  14
1
2
x2  13 x  34
10.1
______________
6.1
3.
1 2
a ( 7a 3  a 2  9a )
3
10.1
&
10.2
Page 6
SUHSD
Standard
Text
Sec.
6.1
10.1
&
10.2
4.
( 3x  7 ) ( 5x 1 )
5.
4x  5y
 x  7y
2
6.1
10.1
2
_____________
6.1
6.
(5x 2  4)  (3x2  8x  4)
10.1
6.1
7.
(5x 2  3xy  y 2 )  (7x 2  8y 2  xy  x3 )
10.1
10.2
10.3
6.1
8.
(2x  y)2
10.1
10.2
10.3
Page 7
SUHSD
Standard
Text
Sec.
(14.8t 2  46.9t)  (10.2t 2  13t)
10.1
&
10.2
6.1
9.
6.1
10. (w2  3w  5) (w2  2w  7)
10.1
&
10.2
6.1
11. Write an expression for the area of a rectangle
10.1
&
10.2
with the following dimensions:
(2x + 3) by (x + 4)
(Hint: Draw a figure.)
6.1
12. The perimeter of a rectangle is 8y + 6, and the
length is 3y + 1. Write an expression for the
width of the rectangle.
10.1
&
10.2
This is an interesting
problem that requires
understanding of perimeter
(Hint: Draw a figure.)
and facility with several
algebraic procedures.
Page 8
SUHSD
Standard
6.1
13. Explain why a  b  a  b  a 2  b 2 . (In other
words, why is there no “middle” term in your
answer when you multiply these two binomials?)
Text
Sec.
10.1
&
10.2
If students have had
opportunities to do similar
explanations during the
unit, then this is an
interesting way to find out
whether they understand
various procedures.
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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:





Performs mathematical operations accurately
Shows logical connections from one step to another.
Completes all problems
Simplifies rational numbers
Expresses answers in standard form
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 10
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.
Opportunities to Learn:
Lesson 1: ADDING POLYNOMIALS and STANDARD FORM
 Terminology: polynomial, standard form, binomial,
monomial, like terms
 Adding polynomials, using both horizontal and vertical
formats. Teach the vertical format alongside the
horizontal.
Lesson 2: SUBTRACTING POLYNOMIALS
 Distributing the negative sign across the entire parentheses
 Subtracting polynomials
Lesson 3: MULTIPLYING A POLYNOMIAL BY A MONOMIAL
 Multiplying a monomial by a binomial
 Multiplying a monomial by a polynomial with three or
more terms
 Alternative models for representing the product:
5
x
y
x+y
= 5x + 5y

=
5
5x + 5y
;
5 (x + y)
= 5x + y
Increase rigor as practice progresses by adjusting
complexity of the monomial expression.
Lesson 4: MULTIPLYING POLYNOMIALS USING AN AREA
MODEL
 Binomial x binomial
 Binomial x trinomial
 Trinomial x trinomial (This is not included in the text; be
sure to include practice in this.)
Lesson 5: MULTIPLYING POLYNOMIALS USING OTHER
METHODS
 Using FOIL
 Relating FOIL to the area model
 Relating FOIL to the distributive property (monomial x
binomial)
 Multiplying Polynomials Vertically
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
Lesson 6: SPECIAL PRODUCTS
 (a+b)2
 (a-b)2
 (a+b)(a-b)
 Special products will be taught as a time-saving activity.
Students will not be penalized for choosing other methods
to (i.e. FOIL) over remembering about special products.
Opportunities to Perform:
Students will have the following opportunities to perform (practice) before
taking the unit assessment:









Practice with adding polynomials
Practice writing answers in standard form
Practice subtracting polynomials
**A short formative assessment covering just addition and
subtraction of polynomials**
Practice with distributing: multiplying a monomial by a
polynomial
Practice with multiplying polynomials using an area model.
**A short formative assessment on multiplying a monomial
by a polynomial and multiplication of polynomials using an
area model.**
Practice with multiplying binomials using FOIL
Practice with using vertical multiplication to multiply
polynomials
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
VI. THE PERFORMANCE STANDARD
Rubric or other form of scoring guide
Meets the Standard
Level 4







Almost all answers correct; all problems attempted.
Consistently able to add, subtract, and multiply polynomials to
include those with fractions and decimals.
Always performs the appropriate operation.
Consistently performed compound, multiple operations properly.
Little or no arithmetic errors.
The majority of answers are simplified and in standard form.
Able to successfully square a binomial.


It does not seem that
students would be held to
high levels of
accomplishment if this
rubric is applied. With this
rubric it might be possible
Level 3


How many is “almost all”?
Mostly all answers correct.
Able to consistently perform all three operations on polynomials on
a regular basis: addition, subtraction, and multiplication.
Almost always performs the appropriate operation.
Arithmetic errors are few in number.
for students with minimal
understandings to be
deemed as meeting the
standard.
How much is mostly?
Does not Meet the Standard
This rubric contains too
Level 2






many characteristics that
Some answers correct.
Conceptual understanding of multiplying polynomials is present.
Able to add only single-variable terms and constants with any
consistency.
Able to apply the distributive property.
Performs the appropriate operation most of the time.
Arithmetic errors are common.

Different people would be
able to rate the same
papers quite differently
through their interpretation
of the terminology or
Level 1


are entirely subjective.
Few answers correct.
Performs operations with variables inconsistently.
Frequent arithmetic errors occur.
depending upon their
moods.
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 13
Instructional Leadership Initiative: Supporting Standards-based Practice
VII. SAMPLES OF STUDENT WORK WITH
COMMENTARY
Commentary – Overview:

We had trouble distinguishing in the rubric between a three and a two
paper. The papers had similar conceptual mistakes, but the two had
more numerous arithmetic errors.

Some answers were in Spanish and needed some translation.

We had difficulty in using absolute and comparative terms in the
rubric.

We incorporated into the rubric the fact that conceptual understanding
is of greater value than correct arithmetic computations.

We noticed that some problems should be written differently. Problem
#13 needs to be re-written in order for students to have to show work.
Problem #12 should include the sketch of a rectangle or should ask
students to draw a rectangle. Change problem #11 to say; write an
expression for the area of a rectangle with length of 2x+3 and width of
x+4.

We have doubts as to whether problem #8 should be included because
it is a special product.

The days that our lessons were done, scheduling of grade level
assemblies may have impacted the students performance.
Special products appear
frequently in application
problems and on
standardized tests.
Traditionally they have
been a major element in
the content of algebra.
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
Original Assessment, Page 1 of 1
Samples of Student Work are based on this version.
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 15
Instructional Leadership Initiative: Supporting Standards-based Practice
Original Assessment, Page 2 of 2
Samples of Student Work are based on this version.
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 16
Instructional Leadership Initiative: Supporting Standards-based Practice
MEETS THE STANDARD:
Performance Level 4
Commentary
This student has
completed all problems
with no conceptual
errors and few if any
arithmetic errors. The
student demonstrates
the ability to
distinguish between
multiplication and
addition of terms
(problem 1), and is
consistently able to
add, subtract, and
multiply. Most
answers are
represented in
standard form(in
descending order of
exponents).
If the requirement is for
the student to state
answers in standard form,
then why would this
student be awarded a
“4”?
Additionally, the
student simplifies
rational expressions
when appropriate
(problems 2 & 3). In
addition and
subtraction of
polynomials, the
student shows
understanding of the
uniqueness of terms by
appropriately creating
placeholders, or “0
terms” when
combining like terms
(problem 6).
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 17
Instructional Leadership Initiative: Supporting Standards-based Practice
MEETS THE STANDARD:
Performance Level 4
Commentary
The student effectively
uses the distributive
property to multiply
binomial factors and is
able to combine like
terms after
multiplication
(problems 4, 8, 10, 11,
& 13). The student
uses his/her own
strategy to account for
and combine like terms
(problem 10). In
subtraction of
polynomials, and
multiplication of
polynomials with
negative terms, the
student demonstrates
accuracy by
distributing the
negative sign
(problems 5, 6, & 13).
The student is able to
understand the zero
sum of the “ab” terms
in (a+b)(a-b) that
generates the
difference of two
squares product
(problem 13).
The student is
successful at
conceptualizing
binomial
representation of
length, width, and
perimeter in a
geometric application
problem, and
accurately computes
the answer (problem
12).
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
MEETS THE STANDARD:
Performance Level 3
Commentary
The student answered
at least 10 of the 13
questions correctly.
Most errors, if any are
arithmetic, not
conceptual.
The student
consistently performs
appropriate operations
on a regular basis.
(see # 1, 2, 3, 4, 5, 7, 9,
10, 11) The student is
able to add, subtract,
and multiply
polynomials.
This student also appears
to only be able to perform
addition of polynomials in
vertical form. Since
flexibility was not part of
the requirements for
meeting the standard, it
does not detract from the
Arithmetic errors are
small and few. For
example, the student
did not reduce 2/4 to
1/2 in #2, and left off
an addition sign in #5.
evidence.
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
MEETS THE STANDARD:
Performance Level 3
Commentary
The student performed
appropriate operation
in all but two problems
(# 8 & 12).
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
Does Not Yet MEET THE STANDARD:
Performance Level 2
Commentary
The student shows
some conceptual
understanding of
multiplying and adding
polynomials (problems
1, 2, 4, 7, 8, 10, 11, &
12) but demonstrates
lack of consistency
with combining
multiple-variable terms
(problems 7 & 8). The
student performs the
appropriate operation
most of the time, yet
arithmetic errors occur
(problem 10), and
rational answers are
not simplified (problem
2).
When using the area
model to multiply, the
student is fairly
accurate (problems 4,
8, 10, & 11), but makes
errors (as in problem
10, where the student
multiplies 2w times 5
and generates 7w).
The student is not able
to multiply a monomial
by a polynomial,
making errors in
multiplying rational
coefficients and
increasing the degree
of the exponent in
terms (problem 3).
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
Does Not Yet MEET THE STANDARD:
Performance Level 2
Commentary
The student lacks the
ability to distribute the
negative throughout a
polynomial when
subtracting (problems
5, 6, & 9), yet correctly
aligns like-terms in
vertical representation
(problems 6, 9, & 12).
The student does not
apply a negative value
to the y terms when
using the area model in
problem 8.
The student is not able
to conceptualize
binomial
representation of
length, width, and
perimeter in a
geometric application
problem, and adds the
binomials in problem
12.
The student is able to
understand the zero
sum of the “ab” terms
in (a+b)(a-b) that
generates the
difference of two
squares product
(problem 13).
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
Does Not Yet MEET THE STANDARD:
Performance Level 1
Commentary
The student is not
consistent with
operations (addition,
subtraction, and
multiplication).
The student does not
combine like terms
correctly.
There are many
arithmetic errors. The
student made many
mistakes with signed
numbers (see #1, 5).
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
Does Not Yet MEET THE STANDARD:
Performance Level 1
Commentary
The student answered
one of the thirteen
problems correctly
(#9). All other
solutions contain many
errors.
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
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Instructional Leadership Initiative: Supporting Standards-based Practice
STUDENT FORMS
Mar Vista Algebra: Adding, Subtracting, and Multiplying Polynomials
Revised 10/30/02
Western Assessment Collaborative at
Page 25