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Algebra 1—An Open Course
Professional Development
Unit 8 – Table of Contents
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Unit 8: Polynomials
Video Overview
Learning Objectives
8.2
Media Run Times
8.3
Instructor Notes
8.4
• The Mathematics of Monomials and Polynomials
• Teaching Tips: Conceptual Challenges and Approaches
• Teaching Tips: Algorithmic Challenges and Approaches
Instructor Overview
8.8
• Tutor Simulation: Roman Numerals and Polynomials
Instructor Overview
• Puzzle: Polynomial Poke
8.9
Instructor Overview
• Project: It's All Fun and Games
8.11
Glossary
8.16
Common Core Standards
8.17
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Monterey Institute for Technology and Education 2011 V1.1
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Algebra 1—An Open Course
Professional Development
Unit 8 – Learning Objectives
Unit 8: Polynomials
Learning Objectives
Lesson 1: Operations on Monomials
Topic 1: Multiplying and Dividing Monomials
Learning Objectives
• Multiply and divide monomials.
Lesson 2: Operations on Polynomials
Topic 1: Polynomials
Learning Objectives
• Identify monomials, binomials and polynomials.
• Write polynomials to describe real world situations.
Topic 2: Adding and Subtracting Polynomials
Learning Objectives
• Add and subtract polynomials.
Topic 3: Multiplying polynomials
Learning Objectives
• Multiply polynomials and collect the like terms of the resulting sum
of monomials.
Topic 4: Special Products of Polynomials
Learning Objectives
• Identify and multiply binomial products.
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Algebra 1—An Open Course
Professional Development
Unit 8 – Media Run Times
Unit 8
Lesson 1
Topic 1, Presentation – 4 minutes
Topic 1, Worked Example 1 – 1.8 minutes
Topic 1, Worked Example 2 – 4 minutes
Topic 1, Worked Example 3 – 2.3 minutes
Lesson 2
Topic 1, Presentation – 3.8 minutes
Topic 1, Worked Example 1 – 4.6 minutes
Topic 1, Worked Example 2 – 1.3 minutes
Topic 2, Presentation – 4.4 minutes
Topic 2, Worked Example 1 – 2 minutes
Topic 2, Worked Example 2 – 1.7 minutes
Topic 2, Worked Example 3 – 2.4 minutes
Topic 3, Presentation 1 – 2.7 minutes
Topic 3, Worked Example 1 – 4.5 minutes
Topic 3, Worked Example 2 – 4.7 minutes
Topic 3, Worked Example 3 – 4.3 minutes
Topic 4, Presentation – 5.3 minutes
Topic 4, Worked Example 1 – 3.3 minutes
Topic 4, Worked Example 2 – 5 minutes
Topic 4, Worked Example 3 – 2.5 minutes
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Algebra 1—An Open Course
Professional Development
Unit 8 – Instructor Notes
Unit 8: Polynomials
Instructor Notes
The Mathematics of Monomials and Polynomials
Unit 8 introduces polynomials and teaches students how to work with them no matter
how many terms they contain (in this course, monomials are included in the definition of
polynomials). Students will learn how to carry out all the basic mathematical operations
on polynomials. They’ll also gain experience writing polynomials from verbal descriptions
of real world situations.
The ability to work fluently with polynomials will be critical for students who progress into
higher math classes like Algebra 2 and beyond.
Teaching Tips: Conceptual Challenges and Approaches
Working with polynomials can present a significant challenge for many students. Most of
the mathematics concerns symbolic manipulation, and if students don’t build a
meaningful conceptual understanding of how and why the techniques work, they will get
lost as the terms become more numerous and complicated.
Using a visual model can be very helpful when working with polynomials. Try beginning
with terms that describe real objects, which can be sketched out and then manipulated
and counted up.
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Algebra 1—An Open Course
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Example
In the presentation for Lesson 2, Topic 3, the concept of multiplying two polynomials is
introduced through the visual of planting a garden.
Students see the result of multiplying polynomials as an understandable collection of
objects instead of just symbols. They can count the various vegetables, compare them
to the original terms, and by doing so, learn to appreciate what actually happens when
polynomials are multiplied.
Hands-On Opportunities
The example above used the area model as a basis for understanding polynomial
multiplication. Students can use virtual algebra tiles for further practice of this technique.
One of the most useful virtual manipulative websites can be found here [MAC users will
need to copy/paste url into browser]:
http://courses.wccnet.edu/~rwhatcher/VAT/SimplifyingPolynomials/
Sketches and manipulatives are powerful tools that can help students build
understanding and practice techniques. However, there are two very important ideas to
keep in mind:
1. Students will need significant guidance to understand manipulatives, especially in
the beginning. We suggest demonstrating any virtual tool in the classroom, either
just by projecting the image and solving the problem on the computer, or better
still, by using an interactive whiteboard and showing students how to solve this
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Algebra 1—An Open Course
Professional Development
problem. Students could also use the board to demonstrate their ideas about
how to take next steps. After seeing it done in the classroom, they’ll be in a
position to work with a tool like this either alone or in small groups.
2. Visual representation tools help students get started, but they are not an
alternative method for ultimately doing the mathematics. Students still need to be
fluent with the relevant symbolic manipulation. It is very important to discuss the
connections between a visual model and its symbolic counterpart when working
with polynomials.
Teaching Tips: Algorithmic Challenges and Approaches
It’s tempting to teach students tricks for memorizing algebraic techniques. A lot of
traditional teaching materials suggest using the acronym FOIL as a mnemonic device to
remember how to multiply two binomials.
While there is nothing inherently wrong with this memorization approach, it does have
limitations. This mnemonic only works when multiplying two binomials—when students
who have grown comfortable with it are confronted by a more challenging situation like
( x + y + 2 ) ( 3 ! 2x ) , they’ll often struggle. As a result, it is more productive to teach a
more general rule from the beginning: “When multiplying two polynomials, multiply
everything in the first parenthesis to everything in the second parenthesis”. That
approach is used exclusively in this course.
If students are struggling to keep track of all of the terms when multiplying polynomials,
they may find it useful to create a rectangular table (which is obviously connected to the
visual area model) to diagram this operation.
Example
( x + y + 2 ) ( 3 ! 2x )
3
!2x
y
3y
!2xy
x
3x
!2x 2
2
6
!4x
Once students have completed the multiplication, they can easily collect the like terms
and find the answer.
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Algebra 1—An Open Course
Professional Development
Summary
This unit focuses on the addition, subtraction, multiplication, and division of polynomials.
It uses general rules and visual models to explain the conceptual basis and the
procedures involved in these operations. Students who struggle with these ideas may
benefit from virtual or hands-on manipulatives, but they must learn how to carry out
strictly symbolic manipulations.
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Algebra 1—An Open Course
Professional Development
Unit 8 – Tutor Simulation
Unit 8: Polynomials
Instructor Overview
Tutor Simulation: Roman Numerals and Polynomials
Purpose
This simulation is designed to challenge a student’s understanding of polynomials.
Students will be asked to apply what they have learned to solve a real world problem by
demonstrating understanding of the following areas:
•
•
•
•
•
•
Polynomials
Multiplying Polynomials
The Distributive Property
The Associative Property
The Commutative Property
Applying Properties to Polynomials
Problem
Students are given the following problem:
You will take a look at Roman numerals and see how working with them is similar to
working with polynomials. Once familiar with Roman numerals, you'll learn how to
multiply them, then apply the same steps to multiply polynomials.
Recommendations
Tutor simulations are designed to give students a chance to assess their understanding
of unit material in a personal, risk-free situation. Before directing students to the
simulation,
•
•
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make sure they have completed all other unit material.
explain the mechanics of tutor simulations
o Students will be given a problem and then guided through its solution by a
video tutor;
o After each answer is chosen, students should wait for tutor feedback
before continuing;
o After the simulation is completed, students will be given an assessment of
their efforts. If areas of concern are found, the students should review unit
materials or seek help from their instructor.
emphasize that this is an exploration, not an exam.
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Algebra 1—An Open Course
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Unit 8 – Puzzle
Unit 8: Polynomials
Instructor Overview
Puzzle: Polynomial Poke
Objective
Polynomial Poke challenges students' familiarity with polynomial nomenclature. To play
the game successfully, they must be able to distinguish between cubic, quadratic, and
linear terms, and recognize monomials, binomials, and trinomials.
Figure 1. Polynomial Poke asks players to pop balloons that contain specified types of
polynomials.
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Algebra 1—An Open Course
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Description
There are three levels in this puzzle, which each consist of 10 groups of floating balloons
containing polynomials. In the first level, learners are challenged to pop balloons in order
of degree of monomials, from cubic to quadratic to linear. In the second level, players
must pop balloons depending on the number of terms in their polynomials. In the third
level, players are asked to pop only those balloons that contain a specified degree of
polynomial. Players earn points for correct answers and lose points for popping balloons
out of sequence.
The puzzle is primarily designed for a single player but in a classroom it could be played
in a group with learners identifying the order or the degree and calling out the balloon for
one to pop.
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Algebra 1—An Open Course
Professional Development
Unit 8 – Project
Unit 8: Polynomials
Instructor Overview
Project: It's All Fun and Games
Student Instructions
Introduction
In algebra variables are used to represent unknowns. When first beginning algebra, the
symbolic representation can be difficult. By now you should be quite comfortable with x
and y, as symbols for unknowns, however, the roots of mathematics are engrained in
complex symbol-based number systems. Get ready to explore these ancient systems
and become an expert at interpreting and using the symbols found within!
Task
Working together with your group, you will research one of four ancient number systems.
Then, based upon what you have learned, you will design a team game based on that
number system. The game needs to be complete with rules, scoring guidelines, and
dimensions of the field based on the ancient number system. Finally, you will calculate
the perimeter of your field using the number system.
Instructions
Solve each problem in order. Save your work along the way, as you will create a
presentation at the conclusion of the project. Your audience will be the Mayan,
Egyptian, Sumerian, or Roman people. You may use multi-media, make a movie, or
create a website to highlight your game and how it connects to the number system.
1
First problem:
•
With your group, choose one of the ancient number systems
below to research. You may use the following links to begin, but
there are multiple websites dedicated to the number systems.
Egyptian:
http://www.touregypt.net/featurestories/numbers.htm
Mayan:
http://www-history.mcs.stand.ac.uk/HistTopics/Mayan_mathematics.html
Roman:
http://turner.faculty.swau.edu/mathematics/materialslibrary/roman/
Sumerian:
http://www.storyofmathematics.com/sumerian.html
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Algebra 1—An Open Course
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Use the following questions to guide your research:
How do you write the following numbers: 1, 5, 10, 20, 50, & 100?
What is the base of the number system?
How has the number system impacted the base ten system that is
used today?
How do you perform basic addition using the number system?
Second problem:
•
Once your group has a good understanding of the number
system, begin thinking about games that are played today and
how they might be adapted to fit with the number system. For
instance, consider football. The field is based on 100 yards and
advancing the ball is based on ten-yard increments. This game
would fit well for the Egyptians and Romans, but not for the
Sumerians and Mayan. Creativity and originality will make your
presentation stand out.
•
A good example of a made-up game is “Quidditch” from the Harry
Potter book series. Information about the fictional game, including
rules and field dimensions can be found at:
http://en.wikipedia.org/wiki/Quidditch
•
First, decide the dimensions of your field. Be sure to keep in mind
the foundation of your number system when making your
decisions. Your dimensions should be written using the symbols
from the system. You can draw a picture of your field using
drafting software such as, Google Sketchup. The free download
is available at http://sketchup.google.com/. If you have an artistic
flair, a hand-drawn field or court is another option.
Third Problem:
•
Now your team needs to work on developing rules and scoring
guidelines for your game. What tools are necessary to play? How
many points are various tasks worth? How many points are
necessary to win? How many players are on the field at once?
Make sure to consider how the answers to each of these
questions would be impacted based on the number system that is
studied.
Hint: (Remember that you will be presenting the game at the end
of the project. It may help to actually go outside and attempt to
play the game in order to discover what works and what does not.)
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Fourth problem:
•
Your final task is to calculate the perimeter of the field, using the
number system studied. Begin by learning to add small numbers
within the system and then work your way to larger numbers. You
will need to include the detailed process that was used to add the
perimeter as part of your presentation.
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Algebra 1—An Open Course
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Collaboration!
Get together with another group to discuss your game and how it is played. Discuss
how the rules and field dimensions relate to what you have studied about the number
system. Finally, work together to check each other’s perimeter calculations. While
reviewing the perimeter calculations, answer the following questions:
•
How is adding within your number system related to adding
polynomials?
•
Do you see any other ties to algebra within your calculations?
Conclusions
Now you will get to present your game using modern technology to your classmates,
who will be considered citizens of the ancient civilization that you researched. Some
options for the presentation include using multi-media, making a movie, or creating a
website to highlight your game and how it connects to the number system. Your
presentation should include answers to each of the four problems above.
Instructor Notes
Assignment Procedures
Problem 1
It is important for students to master basic addition and regrouping within the number
system they have chosen before moving on. By the end, each group will calculate the
perimeter of the playing field. Without a solid understanding of basic addition within their
system, completing more difficult calculations will not be possible.
Problem 4
If a group chooses to make their field LXV by XXV, the calculation for perimeter would
be:
By Addition: LXV + LXV + XXV + XXV
By combining like terms: LLXXXXXXVVVV
By simplifying: LL = C and VVVV = XX
By substitution: CXXXXXXXX
By simplifying: XXXXX=L
By substitution: CLXXX
By showing their work and justifying each step, the students should be able to see how
addition within the number system relates very closely to adding polynomials in algebra.
Recommendations:
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have students work in teams to encourage brainstorming and cooperative
learning.
assign a specific timeline for completion of the project that includes milestone
dates.
provide students feedback as they complete each milestone.
ensure that each member of student groups has a specific job.
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Algebra 1—An Open Course
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Technology Integration
This project provides abundant opportunities for technology integration, and gives
students the chance to research and collaborate using online technology. The students’
instructions list several websites that provide information on numbering systems, game
design, and graphics.
The following are other examples of free Internet resources that can be used to support
this project:
http://www.moodle.org
An Open Source Course Management System (CMS), also known as a Learning
Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has
become very popular among educators around the world as a tool for creating online
dynamic websites for their students.
http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview
Allows you create a secure online Wiki workspace in about 60 seconds. Encourage
classroom participation with interactive Wiki pages that students can view and edit from
any computer. Share class resources and completed student work with parents.
http://www.docs.google.com
Allows students to collaborate in real-time from any computer. Google Docs provides
free access and storage for word processing, spreadsheets, presentations, and surveys.
This is ideal for group projects.
http://why.openoffice.org/
The leading open-source office software suite for word processing, spreadsheets,
presentations, graphics, databases and more. It can read and write files from other
common office software packages like Microsoft Word or Excel and MacWorks. It can be
downloaded and used completely free of charge for any purpose.
Rubric
Score
4
Content
Your project appropriately answers
each of the problems. Background
research is thorough. A detailed
drawing of the game field is given,
with dimensions labeled using the
number system.
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3
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Rules and scoring guidelines are
complete and relate to the number
system studied. A detailed
calculation of perimeter, in the
number system, is included.
Your project appropriately answers
each of the problems. Background
research is thorough. A detailed
drawing of the game field is given,
with dimensions labeled using the
Presentation
Your project contains information presented in
a logical and interesting sequence that is easy
to follow.
Your project is professional looking with
graphics and attractive use of color.
Your project contains information presented in
a logical sequence that is easy to follow.
Your project is neat with graphics and
attractive use of color.
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Algebra 1—An Open Course
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number system. Minor errors may
be noted.
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2
Rules and scoring guidelines are
complete and relate to the number
system studied. A detailed
calculation of perimeter, in the
number system, is included. Minor
errors may be noted.
Your project attempts to answer
each of the problems. Background
research is present, but not
complete. A drawing of the game
field is given, with dimensions
labeled using the number system.
Major errors may be noted &/or
some information is missing.
Your project is hard to follow because the
material is presented in a manner that jumps
around between unconnected topics.
Your project contains low quality graphics and
colors that do not add interest to the project.
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1
Rules and scoring guidelines
present and relate to the number
system studied. The perimeter is
given, but the detailed work used to
obtain the answer is not given.
Major errors may also be noted.!
Your project attempts to answer
some of the problems. Major errors
are noted and information is missing.
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Your project has minimal information
on rules and scoring guidelines. The
perimeter calculation is missing. !
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Your project is difficult to understand because
there is no sequence of information.
Your project is missing graphics and uses
little to no color.
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Algebra 1—An Open Course
Professional Development
Unit 8 – Glossary
Glossary
Unit 8: Algebra - Polynomials
area model
a graphic representation of a multiplication problem, in which the
length and width of a rectangle are the factors and the area is the
product
binomial
a sum of two monomials, such as 3x + 7
coefficient
a number that multiplies a variable
like terms
two or more monomials that contain the same variables raised to
the same powers, regardless of their coefficients. For example,
2x2y and -8x2y are like terms because they have the same
variables raised to the same exponents.
monomial
a number, a variable, or a product of a number and one or more
variables with whole number exponents, such as -5, x, and 8xy3
polynomial
a monomial or sum of monomials, like 4x2 + 3x – 10
special product
a product resulting from binomial multiplication that has certain
characteristics. For example x2 – 25 is called a special product
because both its terms are perfect squares and it can be factored
into (x + 5)(x– 5).
term
a value in a sequence--the first value in a sequence is the 1st term,
the second value is the 2nd term, and so on; a term is also any of
the monomials that make up a polynomial
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Algebra 1—An Open Course
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Unit 8 – Common Core
NROC Algebra 1--An Open Course
Unit 8
Mapped to Common Core State Standards, Mathematics
Algebra 1 | Polynomials | Operations on Monomials | Multiplying and Dividing Monomials
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-SSE.
Seeing Structure in
Expressions
Interpret the structure
of expressions.
Use the structure of an
expression to identify
ways to rewrite it. For
example, see x^4 - y^4
as (x^2)^2 - (y^2)^2,
thus recognizing it as a
difference of squares
that can be factored as
(x^2 - y^2)(x^2 + y^2).
STANDARD
EXPECTATION
A-SSE.2.
Algebra 1 | Polynomials | Operations on Polynomials | Polynomials
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-SSE.
EXPECTATION
A-SSE.1.
GRADE EXPECTATION
A-SSE.1.a.
STRAND / DOMAIN
CC.A.
Seeing Structure in
Expressions
Interpret the structure
of expressions.
Interpret expressions
that represent a
quantity in terms of its
context.
Interpret parts of an
expression, such as
terms, factors, and
coefficients.
Algebra
CATEGORY / CLUSTER
A-CED.
Creating Equations
STANDARD
STANDARD
!
EXPECTATION
A-CED.2.
EXPECTATION
A-CED.3.
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Create equations that
describe numbers or
relationships.
Create equations in two
or more variables to
represent relationships
between quantities;
graph equations on
coordinate axes with
labels and scales.
Represent constraints by
equations or
inequalities, and by
systems of equations
and/or inequalities, and
interpret solutions as
viable or nonviable
options in a modeling
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Algebra 1—An Open Course
Professional Development
context. For example,
represent inequalities
describing nutritional
and cost constraints on
combinations of
different foods.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-BF.
Building Functions
STANDARD
EXPECTATION
F-BF.1.
GRADE EXPECTATION
F-BF.1.a.
Build a function that
models a relationship
between two quantities.
Write a function that
describes a relationship
between two quantities.
Determine an explicit
expression, a recursive
process, or steps for
calculation from a
context.
Algebra 1 | Polynomials | Operations on Polynomials | Adding and Subtracting Polynomials
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-SSE.
Seeing Structure in
Expressions
Interpret the structure
of expressions.
Interpret expressions
that represent a
quantity in terms of its
context.
Interpret parts of an
expression, such as
terms, factors, and
coefficients.
Algebra
STANDARD
EXPECTATION
A-SSE.1.
GRADE EXPECTATION
A-SSE.1.a.
STRAND / DOMAIN
CC.A.
CATEGORY / CLUSTER
A-SSE.
STANDARD
EXPECTATION
A-SSE.2.
STRAND / DOMAIN
CC.A.
CATEGORY / CLUSTER
A-APR.
STANDARD
EXPECTATION
!
A-APR.1.
"#$"!
Seeing Structure in
Expressions
Interpret the structure
of expressions.
Use the structure of an
expression to identify
ways to rewrite it. For
example, see x^4 - y^4
as (x^2)^2 - (y^2)^2,
thus recognizing it as a
difference of squares
that can be factored as
(x^2 - y^2)(x^2 + y^2).
Algebra
Arithmetic with
Polynomials and
Rational Functions
Perform arithmetic
operations on
polynomials.
Understand that
polynomials form a
system analogous to the
integers, namely, they
are closed under the
operations of addition,
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Algebra 1—An Open Course
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subtraction, and
multiplication; add,
subtract, and multiply
polynomials.
Algebra 1 | Polynomials | Operations on Polynomials | Multiplying Polynomials
Grade: 7 - Adopted 2010
STRAND / DOMAIN
CC.7.EE.
CATEGORY / CLUSTER
STANDARD
7.EE.1.
Expressions and
Equations
Use properties of
operations to generate
equivalent expressions.
Apply properties of
operations as strategies
to add, subtract, factor,
and expand linear
expressions with rational
coefficients.
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-SSE.
Seeing Structure in
Expressions
Interpret the structure
of expressions.
Interpret expressions
that represent a
quantity in terms of its
context.
Interpret parts of an
expression, such as
terms, factors, and
coefficients.
Algebra
STANDARD
EXPECTATION
A-SSE.1.
GRADE EXPECTATION
A-SSE.1.a.
STRAND / DOMAIN
CC.A.
CATEGORY / CLUSTER
A-SSE.
STANDARD
EXPECTATION
A-SSE.2.
STRAND / DOMAIN
CC.A.
CATEGORY / CLUSTER
A-APR.
STANDARD
EXPECTATION
!
A-APR.1.
"#$+!
Seeing Structure in
Expressions
Interpret the structure
of expressions.
Use the structure of an
expression to identify
ways to rewrite it. For
example, see x^4 - y^4
as (x^2)^2 - (y^2)^2,
thus recognizing it as a
difference of squares
that can be factored as
(x^2 - y^2)(x^2 + y^2).
Algebra
Arithmetic with
Polynomials and
Rational Functions
Perform arithmetic
operations on
polynomials.
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.
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!
Algebra 1—An Open Course
Professional Development
Algebra 1 | Polynomials | Operations on Polynomials | Special Products of Polynomials
Grade: 7 - Adopted 2010
STRAND / DOMAIN
CC.7.EE.
CATEGORY / CLUSTER
STANDARD
7.EE.1.
Expressions and
Equations
Use properties of
operations to generate
equivalent expressions.
Apply properties of
operations as strategies
to add, subtract, factor,
and expand linear
expressions with rational
coefficients.
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-SSE.
Seeing Structure in
Expressions
Interpret the structure
of expressions.
Interpret expressions
that represent a
quantity in terms of its
context.
Interpret parts of an
expression, such as
terms, factors, and
coefficients.
Algebra
STANDARD
EXPECTATION
A-SSE.1.
GRADE EXPECTATION
A-SSE.1.a.
STRAND / DOMAIN
CC.A.
CATEGORY / CLUSTER
A-APR.
STANDARD
EXPECTATION
A-APR.1.
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"#%,!
Arithmetic with
Polynomials and
Rational Functions
Perform arithmetic
operations on
polynomials.
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.