Week 10: Reading Discussion
Read Standards Deconstructed High School Algebra (in the Course Materials Module) – pp. 15-28.
Respond to the following questions:
HS
COMMON
CORE
MATHEMATICS
855.809.7018 | www.commoncoreinstitute.com
HIGH SCHOOL
ALGEBRA
DECONSTRUCTED for
CLASSROOM IMPACT
State Standards
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 3
MATHEMATICS
Introduction
The Common Core Institute is pleased to offer this grade-level tool for educators who are teaching with the
Common Core State Standards.
The Common Core Standards Deconstructed for Classroom Impact is designed for educators by educators
as a two-pronged resource and tool 1) to help educators increase their depth of understanding of the
Common Core Standards and 2) to enable teachers to plan College & Career Ready curriculum and
classroom instruction that promotes inquiry and higher levels of cognitive demand.
What we have done is not all new. This work is a purposeful and thoughtful compilation of preexisting
materials in the public domain, state department of education websites, and original work by the Center
for College & Career Readiness. Among the works that have been compiled and/or referenced are the
following: Common Core State Standards for Mathematics and the Appendix from the Common Core State
Standards Initiative; Learning Progressions from The University of Arizona’s Institute for Mathematics and
Education, chaired by Dr. William McCallum; the Arizona Academic Content Standards; the North Carolina
Instructional Support Tools; and numerous math practitioners currently in the classroom.
We hope you will find the concentrated and consolidated resource of value in your own planning. We also
hope you will use this resource to facilitate discussion with your colleagues and, perhaps, as a lever to help
assess targeted professional learning opportunities.
Understanding the Organization
The Overview acts as a quick-reference table of contents
as it shows you each of the domains and related clusters
covered in this specific grade-level booklet. This can help
serve as a reminder of what clusters are part of which
domains and can reinforce the specific domains for each
grade level.
Critical Areas of Focus is designed to help you begin to
approach how to examine your curriculum, resources,
and instructional practices. A review of the Critical Areas
of Focus might enable you to target specific areas of
professional learning to refresh, as needed.
For each domain is the domain itself and the associated
clusters. Within each domain are sections for each of the
associated clusters. The cluster-specific content can take you
to a deeper level of understanding. Perhaps most importantly, we include here the Learning Progressions.
The Learning Progressions provide context for the current domain and its related standards. For any grade
except Kindergarten, you will see the domain specific standards for the current grade in the center column.
To the left are the domain-specific standards for the preceding grade and to the right are the domain-
specific standards for the following grade. Combined with the Critical Areas of Focus, these Learning
Progressions can assist you in focusing your planning.
Math Fluency Standards
K Add/subtract within 5
1 Add/subtract within 10
2
Add/subtract within 20
Add/subtract within 100 (pencil & paper)
3
Multiply/divide within 100
Add/subtract within 1000
4 Add/subtract within 1,000,000
5 Multi-digit multiplication
6
Multi-digit division
Multi-digit decimal operations
7 Solve px + q = r, p(x + q) = r
8 Solve simple 2 x 2 systems by inspection
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4 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
For each cluster, we have included four key sections: Description, Big Idea, Academic Vocabulary, and
Deconstructed Standard.
The cluster Description offers clarifying information, but also points to the Big Idea that can help you focus
on that which is most important for this cluster within this domain. The Academic Vocabulary is derived
from the cluster description and serves to remind you of potential challenges or barriers for your students.
Each standard specific to that cluster has been deconstructed. There Deconstructed Standard for each
standard specific to that cluster and each Deconstructed Standard has its own subsections, which can
provide you with additional guidance and insight as you plan. Note the deconstruction drills down to the
sub-standards when appropriate. These subsections are:
� Standard Statement
� Standard Description
� Essential Question(s)
� Mathematical Practice(s)
� DOK Range Target for Learning and Assessment
� Learning Expectations
� Explanations and Examples
As noted, first are the Standard Statement and Standard Description, which are followed by the Essential
Question(s) and the associated Mathematical Practices. The Essential Question(s) amplify the Big Idea,
with the intent of taking you to a deeper level of understanding; they may also provide additional context
for the Academic Vocabulary.
The DOK Range Target for Learning and Assessment remind you of the targeted level of cognitive
demand. The Learning Expectations correlate to the DOK and express the student learning targets for
student proficiency for KNOW, THINK, and DO, as appropriate. In some instances, there may be no learning
targets for student proficiency for one or more of KNOW, THINK or DO. The learning targets are expressions
of the deconstruction of the Standard as well as the alignment of the DOK with appropriate consideration of
the Essential Questions.
The last subsection of the Deconstructed Standard includes Explanations and Examples. This subsection
might be quite lengthy as it can include additional context for the standard itself as well as examples of
what student work and student learning could look like. Explanations and Examples may offers ideas for
instructional practice and lesson plans.
A wonderful resource for explanations and examples, which we often referred to and cited as a source in this
tool, is www.shmooop.com.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 5
MATHEMATICS
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PRACTICE EXPLANATION
MP.1 Make sense
and persevere in
problem solving.
Mathematically proficient students start by explaining to themselves the meaning
of a problem and looking for entry points to its solution. They analyze givens,
constraints, relationships, and goals. They make conjectures about the form and
meaning of the solution and plan a solution pathway rather than simply jumping
into a solution attempt. They consider analogous problems, and try special cases
and simpler forms of the original problem in order to gain insight into its solution.
They monitor and evaluate their progress and change course if necessary. Older
students might, depending on the context of the problem, transform algebraic
expressions or change the viewing window on their graphing calculator to get
the information they need.
Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important
features and relationships, graph data, and search for regularity or trends. Younger
students might rely on using concrete objects or pictures to help conceptualize
and solve a problem. Mathematically proficient students check their answers to
problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving
complex problems and identify correspondences between different approaches.
MP.2 Reason
abstractly and
quantitatively.
Mathematically proficient students make sense of the quantities and their
relationships in problem situations. Students bring two complementary
abilities to bear on problems involving quantitative relationships: the ability to
decontextualize—to abstract a given situation and represent it symbolically and
manipulate the representing symbols as if they have a life of their own, without
necessarily attending to their referents—and the ability to contextualize, to pause
as needed during the manipulation process in order to probe into the referents
for the symbols involved. Quantitative reasoning entails habits of creating a
coherent representation of the problem at hand; considering the units involved;
attending to the meaning of quantities, not just how to compute them; and
knowing and flexibly using different properties of operations and objects.
Standards for Mathematical Practice in High School
Mathematics Courses
The Standards for Mathematical Practice describe varieties of expertise that
mathematics educators at all levels should seek to develop in their students. These
practices rest on important “processes and proficiencies” with longstanding
importance in mathematics education. The first of these are the NCTM process
standards of problem solving, reasoning and proof, communication, representation,
and connections. The second are the strands of mathematical proficiency specified
in the National Research Council’s report Adding It Up: adaptive reasoning, strategic
competence, conceptual understanding (comprehension of mathematical concepts,
operations and relations), procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently, and appropriately), and productive disposition (habitual
inclination to see mathematics as sensible, useful, and worthwhile, coupled with a
belief in diligence and one’s own efficacy).
6 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
PRACTICE EXPLANATION
MP.3 Construct
viable arguments
and critique the
reasoning of
others.
Mathematically proficient students understand and use stated assumptions,
definitions, and previously established results in constructing arguments. They
make conjectures and build a logical progression of statements to explore
the truth of their conjectures. They are able to analyze situations by breaking
them into cases, and can recognize and use counterexamples. They justify their
conclusions, communicate them to others, and respond to the arguments of
others.
They reason inductively about data, making plausible arguments that take
into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments,
distinguish correct logic or reasoning from that which is flawed, and—if there is a
flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as
objects, drawings, diagrams, and actions. Such arguments can make sense and be
correct, even though they are not generalized or made formal until later grades.
Later, students learn to determine domains to which an argument applies.
Students at all grades can listen or read the arguments of others, decide whether
they make sense, and ask useful questions to clarify or improve the arguments.
MP.4 Model with
mathematics.
Mathematically proficient students can apply the mathematics they know to
solve problems arising in everyday life, society, and the workplace. In early grades,
this might be as simple as writing an addition equation to describe a situation.
In middle grades, a student might apply proportional reasoning to plan a school
event or analyze a problem in the community. By high school, a student might
use geometry to solve a design problem or use a function to describe how one
quantity of interest depends on another. Mathematically proficient students
who can apply what they know are comfortable making assumptions and
approximations to simplify a complicated situation, realizing that these may need
revision later.
They are able to identify important quantities in a practical situation and map
their relationships using such tools as diagrams, 2-by-2 tables, graphs, flowcharts
and formulas. They can analyze those relationships mathematically to draw
conclusions.
They routinely interpret their mathematical results in the context of the situation
and reflect on whether the results make sense, possibly improving the model if it
has not served its purpose.
MP.5 Use
appropriate
tools
strategically.
Mathematically proficient students consider the available tools when solving a
mathematical problem. These tools might include pencil and paper, concrete
models, ruler, protractor, calculator, spreadsheet, computer algebra system,
statistical package, or dynamic geometry software. Proficient students are
sufficiently familiar with tools appropriate for their grade or course to make sound
decisions about when each of these tools might be helpful, recognizing both the
insight to be gained and their limitations. For example, mathematically proficient
high school students interpret graphs of functions and solutions generated using
a graphing calculator.
They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions,
explore consequences, and compare predictions with data. Mathematically
proficient students at various grade levels are able to identify relevant external
mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to
explore and deepen their understanding of concepts.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 7
MATHEMATICS
PRACTICE EXPLANATION
MP.6 Attend to
precision.
Mathematically proficient students try to communicate precisely to others. They
try to use clear definitions in discussion with others and in their own reasoning.
They state the meaning of the symbols they choose, are careful about specifying
units of measure, and labeling axes to clarify the correspondence with quantities
in a problem.
They express numerical answers with a degree of precision appropriate for the
problem context. In the elementary grades, students give carefully formulated
explanations to each other.
By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
MP.7 Look for
and make use of
structure.
Mathematically proficient students look closely to discern a pattern or structure.
Young students, for example, might notice that three and seven more is the
same amount as seven and three more, or they may sort a collection of shapes
according to how many sides the shapes have.
Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in
preparation for learning about the distributive property. In the expression x2 + 9x
+ 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7.
They recognize the significance of an existing line in a geometric figure and can
use the strategy of drawing an auxiliary line for solving problems.
They also can step back for an overview and shift perspective.
They can see complicated things, such as some algebraic expressions, as single
objects or as composed of several objects. For example, they can see 5 – 3(x – y)2
as 5 minus a positive number times a square and use that to realize that its value
cannot be more than 5 for any real numbers x and y.
MP.8 Look for
and express
regularity
in repeated
reasoning.
Mathematically proficient students notice if calculations are repeated, and look
both for general methods and for shortcuts. Upper elementary students might
notice when dividing 25 by 11 that they are repeating the same calculations over
and over again, and conclude they have a repeating decimal. By paying attention
to the calculation of slope as they repeatedly check whether points are on the line
through (1, 2) with slope 3, middle school students might abstract the equation
(y – 2)(x – 1) = 3.
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8 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
OVERVIEW
Expressions
An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations,
exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of
parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that
describes a computation involving a general quantity requires the ability to express the computation in general
terms, abstracting from specific instances.
Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a
different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For
example, p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows
that adding a tax is the same as multiplying the price by a constant factor.
Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of
algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example,
p + 0.05p is the sum of the simpler expressions p and 0.05p. Viewing an expression as the result of operation on
simpler expressions can sometimes clarify its underlying structure.
A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform
complicated algebraic manipulations, and understand how algebraic manipulations behave.
Equations and Inequalities
An equation is a statement of equality between two expressions, often viewed as a question asking for which values
of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An
identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression
in an equivalent form.
The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables
form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or
inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.
An equation can often be solved by successively deducing from it one or more simpler equations. For example,
one can add the same constant to both sides without changing the solutions, but squaring both sides might lead
to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and
anticipating the nature and number of solutions.
Some equations have no solutions in a given number system, but have a solution in a larger system. For example,
the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2x + 1 = 0 is a rational number, not
an integer; the solutions of x2 – 2 = 0 are real numbers, not rational numbers; and the solutions of x2 + 2 = 0 are
complex numbers, not real numbers.
The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula
for the area of a trapezoid, A = ((b1+b2)/2)h, can be solved for h using the same deductive process.
Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of
equality continue to hold for inequalities and can be useful in solving them.
Connections to Functions and Modeling
Expressions can define functions, and equivalent expressions define the same function. Asking when two functions
have the same value for the same input leads to an equation; graphing the two functions allows for finding
approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of
these is an essential skill in modeling.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 9
MATHEMATICS
OVERVIEW
Seeing Structure in Expressions (A-SSE)
� Interpret the structure of expressions.
� Write expressions in equivalent forms to solve problems.
Arithmetic with Polynomials and Rational Expressions (A-APR)
� Perform arithmetic operations on polynomials.
� Understand the relationship between zeros and factors of polynomials.
� Use polynomial identities to solve problems.
� Rewrite rational expressions.
Creating Equations (A-CED)
� Create equations that describe numbers or relationships.
Reasoning with Equations and Inequalities (A-REI)
� Understand solving equations as a process of reasoning and explain the reasoning.
� Solve equations and inequalities in one variable.
� Solve systems of equations.
� Represent and solve equations and inequalities graphically.
Mathematical Practices (MP)
MP 1. Make sense of problems and persevere in solving them.
MP 2. Reason abstractly and quantitatively.
MP 3. Construct viable arguments and critique the reasoning of others.
MP 4. Model with mathematics.
MP 5. Use appropriate tools strategically.
MP 6. Attend to precision.
MP 7. Look for and make use of structure.
MP 8. Look for and express regularity in repeated reasoning.
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The high school standards specify the mathematics that all students should study in order to be college and career
ready. Additional mathematics that students should learn in order to take advanced courses such as calculus,
advanced statistics, or discrete mathematics is indicated by (+), as in this example:
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and
imaginary numbers).
All standards without a (+) symbol should be in the common mathematics curriculum for all college and career
ready students. Standards with a (+) symbol may also appear in courses intended for all students.
10 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
Domain HS Algebra I Mathematics I Geometry
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The Real Number System (RN) RN. 1, 2, 3
Quantities (Q) Q. 1, 2, 3 Q. 1, 2, 3
The Complex Number System (CN)
Vector Quantities and Matrices (VM)
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Seeing Structure in Expressions (SSEE) SSE. 1a, 1b, 2, 3a, 3b, 3c SSE. 1a, 1b
Arithmetic with Polynomials and
Rational Expressions (APR) APR. 1
Creating Equations (CED) CED. 1, 2, 3, 4 CED. 1, 2, 3, 4
Reasoning with Equations and
Inequalities (REI)
REI. 1, 3, 4a, 4b, 5, 6, 7,
10, 11, 12 REI. 1, 3, 5, 6, 10, 11, 12
Fu
nc
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Interpreting Functions (IF) IF. 1, 2, 3, 4, 5, 6, 7a, 7b,
7c, 8a, 8b, 9
IF. 1, 2, 3, 4, 5, 6, 7a,
7c, 9
Building Functions (BF) BF. 1a, 1b, 2, 3, 4a BF. 1a, 1b, 2, 3
Linear, Quadratic, and Exponential
Models (LE) LE. 1a, 1b, 1c, 2, 3, 5 LE. 1a, 1b, 1c, 2, 3, 5
Trigonometric Functions (TF)
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Congruence (CO) CO. 1, 2, 3, 4, 5, 6, 7, 8,
12, 13 CO. 1-13
Similarity, Right Triangles, and
Trigonometry (SRT) SRT. 1-11
Circle (C) C. 1-5
Expressing Geometric Properties with
Equations (GPE) GPE. 4, 5, 7 GPE. 1, 2, 4-7
Geometric Measurement and
Dimension (GMD) GMD. 1, 3, 4
Modeling with Geometry (MG) MG. 1, 2, 3
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Pr
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Interpreting Categorical and
Quantitative Data (ID) ID. 1-3, 5-9 ID. 1-3, 5-9
Making Interference and Justifying
Conclusions (IC)
Conditional Probability and the Rules
of Probability (CP) CP. 1-9
Using Probability to Make Decisions
(MD) MD. 6-7
High School Pathways for Traditional and Integrated Courses
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 11
MATHEMATICS
Mathematics II HS Algebra II Mathematics III 4th Courses (T) 4th Courses (I)
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RN, 1, 2, 3
CN. 1, 2, 7, 8, 9 CN. 1, 2, 7, 8, 9 CN. 8, 9 CN. 3, 4, 5, 6 CN. 3, 4, 5, 6
VM. 1, 2, 3, 4 (a-c), 5(a-
b), 6, 7, 8, 9, 10, 11, 12
VM. 1, 2, 3, 4 (a-c), 5(a-
b), 6, 7, 8, 9, 10, 11, 12
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SSE.1a, 1b, 2, 3a, 3b, 3c SSE. 1, 1b, 2, 4 SSE. 1, 1b, 2, 4
APR. 1 APR. 1-7 APR. 1-7
CED. 1, 2, 4 CED. 1, 2, 3, 4 CED. 1, 2, 4
REI. 4a, 4b, 7 REI. 2, 11 REI. 2, 11 REI. 8, 9 REI. 8, 9
Fu
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IF. 4, 5, 6, 7a, 7b, 8a,
8b, 9 IF. 4, 5, 6, 7b, 7c, 7e, 8, 9 IF. 4, 5, 6, 7b, 7c, 7e, 8, 9 IF. 7d IF. 7d
BF. 1a, 1b, 3, 4a BF. 1b, 3, 4a BF. 1b, 3, 4a BF. 1c, 4b, 4c, 4d, 5 BF. 1c, 4b, 4c, 4d, 5
LE. 3 LE. 4 LE. 4
TF. 8 TF. 1, 2, 5, 8 TF. 1, 2, 5 TF. 3, 4, 6, 7, 9 TF. 3, 4, 6, 7, 9
G
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CO. 9, 10, 11
SRT. 1a, 1b, 2, 3, 4, 5,
6, 7, 8 SRT. 9, 10, 11
C. 1-5
GPE. 1, 2, 4 GPE. 3 GPE. 3
GMD. 1, 3 GMD. 4 GMD. 2 GMD. 2
MG. 1, 2, 3
St
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Pr
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ID. 4 ID. 4
IC. 1-6 IC. 1-6
CP. 1-9
MD. 6-7 MD. 6, 7 MD. 6-7 MD. 1-5 MD. 1-5
High School Pathways for Traditional and Integrated Courses O
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12 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
Algebra
Math 1 Math 2 Math 3 Math 4
Seeing Structure in Expressions (SSE)
A.SSE.1(a-b) A.SSE.1(a-b) A.SSE.1(a-b)
A.SSE.2 A.SSE.2
A.SSE.3
A.SSE.4
Arithmetic with Polynomials and Rational Expressions (APR)
A.APR.1 A.APR.1
A.APR.2
A.APR.3
A.APR.4
A.APR.5
A.APR.6
A.APR.7
Creating Equations (CED)
A.CED.1 A.CED.1 A.CED.1
A.CED.2 A.CED.2 A.CED.2
A.CED.3 A.CED.3 A.CED.3
A.CED.4 A.CED.4 A.CED.4
Reasoning with Equations and Inequalities (REI)
A.REI.1
A.REI.2
A.REI.3
A.REI.4(a-b)
A.REI.5
A.REI.6
A.REI.7
A.REI.8
A.REI.10 A.REI.9
A.REI.11 A.REI.11
A.REI.12
Learning Progressions for Integrated Courses
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 13
MATHEMATICS
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Algebra
Algebra 1 (Grade 9) Geometry (Grade 10) Algebra 2 (Grade 11) PreCalculus (Grade 12)
Seeing Structure in Expressions (SSE)
A.SSE.1(a-b) A.SSE.1(a-b)
A.SSE.2 A.SSE.2
A.SSE.3(a-c)
A.SSE.4
Arithmetic with Polynomials and Rational Expressions (APR)
A.APR.1 A.APR.1
A.APR.2
A.APR.3
A.APR.4
A.APR.5
A.APR.6
A.APR.7
Creating Equations (CED)
A.CED.1 A.CED.1
A.CED.2 A.CED.2
A.CED.3 A.CED.3
A.CED.4 A.CED.4
Reasoning with Equations and Inequalities (REI)
A.REI.1
A.REI.2
A.REI.3
A.REI.4(a-b)
A.REI.5
A.REI.6
A.REI.7
A.REI.8
A.REI.9
A.REI.10
A.REI.11 A.REI.11
A.REI.12
Learning Progressions for Traditional Courses
MATHEMATICS
SEEING STRUCTURE
IN EXPRESSIONS
(A-SSE)
DOMAIN:
HIGH SCHOOL
ALGEBRA
16 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
DOMAIN Seeing Structure in Expressions (A-SSE)
CLUSTERS
1. Interpret the structure of expressions
2. Write expressions in equivalent forms to solve problems
ACADEMIC
VOCABULARY
Algebra I
absolute value, complement of an event, compound, conjunction, direct and inverse variation, disjunction,
domain & range, exponential growth (and decay), interest (simple and compound), irrational numbers, joint and
conditional probability, law of large numbers, mathematical model, measure of spread (range, interquartile range),
midpoint formula, outlier, parent function, Pascal’s triangle, polynomial (binomial, trinomial), quadratic formula
(including discriminant), quantitative and qualitative data, radicand, rational expression, real number properties,
real roots (zeros, solutions, x-intercepts), relative frequency, sequences (arithmetic, geometric, Fibonacci),
simulations, subsets of real numbers
Algebra II
amplitude, asymptote, binomial theorem, combination, common ratio (geometric sequence), complete the
square, complex conjugate, complex number, composition (of functions), conic sections (circles, parabola, ellipse,
hyperbola), empirical rule, factorial, focus (pl. foci), independent and dependent events, inverse of a relation,
logarithm, normal distribution, period, permutation, piece-wise function, radian measure, rational function,
regression equation, series (arithmetic, geometric, finite, infinite, etc.), sigma, standard deviation, step function,
synthetic division, transcendental function, trigonometric function, trigonometric identity, unit circle, variance
CLUSTER 1. Interpret the structure of expressions
BIG IDEA
• Expressions are used to show a mathematical relationship and to designate value.
• Variables, expressions, and equations are algebraic representations of mathematical situations that
dictate the unknown to be solved in real-world problems.
• Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
STANDARD AND DECONSTRUCTION
A.SSE.1 Interpret expressions that represent a quantity in terms of its context. («)
DESCRIPTION
A.SSE.1a Identify the different parts of the expression and explain their meaning within the context of a problem.
A.SSE.1b Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of
the individual parts.
ESSENTIAL
QUESTION(S)
• How can the understanding of the parts of an expression lead to effective problem solving?
• How are terms, factors and coefficients related?
MATHEMATICAL
PRACTICE(S)
HS.MP.1. Make sense of problems and persevere in solving them.
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
SUBSTANDARD
DECONSTRUCTED
A.SSE.1a Interpret parts of an expression, such as terms, factors, and
coefficients.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Define and recognize parts of an
expression, such as terms, factors,
and coefficients.
Interpret parts of an expression,
such as terms, factors, and
coefficients in terms of the context.
SUBSTANDARD
DECONSTRUCTED
A.SSE.1b Interpret complicated expressions by viewing one or more of
their parts as a single entity. For example, interpret P(1+r)n as the product of
P and a factor not depending on P.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Interpret complicated expressions,
in terms of the context, by viewing
one or more of their parts as a
single entity.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 17
MATHEMATICS
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STANDARD AND DECONSTRUCTION
A.SSE.1 Interpret expressions that represent a quantity in terms of its context. («)
DESCRIPTION
A.SSE.1a Identify the different parts of the expression and explain their meaning within the context of a problem.
A.SSE.1b Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of
the individual parts.
ESSENTIAL
QUESTION(S)
• How can the understanding of the parts of an expression lead to effective problem solving?
• How are terms, factors and coefficients related?
MATHEMATICAL
PRACTICE(S)
HS.MP.1. Make sense of problems and persevere in solving them.
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
SUBSTANDARD
DECONSTRUCTED
A.SSE.1a Interpret parts of an expression, such as terms, factors, and
coefficients.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Define and recognize parts of an
expression, such as terms, factors,
and coefficients.
Interpret parts of an expression,
such as terms, factors, and
coefficients in terms of the context.
SUBSTANDARD
DECONSTRUCTED
A.SSE.1b Interpret complicated expressions by viewing one or more of
their parts as a single entity. For example, interpret P(1+r)n as the product of
P and a factor not depending on P.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Interpret complicated expressions,
in terms of the context, by viewing
one or more of their parts as a
single entity.
EXPLANATIONS
AND EXAMPLES
Students should understand the vocabulary for the parts that make up the whole expression and be able to
identify those parts and interpret their meaning in terms of a context.
The only way that could be more general is if it said, “Do things to things in terms of other things.” Luckily, we have
just a smidgeon more to work with.
At its core, this standard wants students to start thinking of math as a language, not a pile of numbers. Just like
any other language, math can help us communicate thoughts and ideas with each other, but students need to
know the basics before they can really understand it.
Your students probably already have some idea of what an expression is in a general sense. Start from this point.
At its simplest, an expression is a thought or idea communicated by language. In the same way, a mathematical
expression can be considered a mathematical thought or idea communicated by the language of mathematics.
Emphasize that mathematics is a language, just as English, French, German, and Pig Latin are languages. Students
should use the ocabulary-vay of athematics-may correctly to become fluent in it. After all, the best way to learn a
new language is to immerse yourself in it.
18 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• What strategies can be applied to create equivalent expressions?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures. Tasks assessing expressing mathematical reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Identify ways to rewrite
expressions, such as difference
of squares, factoring out a
common monomial, and
regrouping.
Identify various structures of
expressions.
Use the structure of an expression to
identify ways to rewrite it.
Classify expressions by structure
and develop strategies to assist in
classification.
EXPLANATIONS
AND EXAMPLES
Students should extract the greatest common factor (whether a constant, a variable, or a combination of each). If
the remaining expression is quadratic, students should factor the expression further.
In English, we have many different ways of saying the same thing. “Stop that!” and “Stop eating that!” and “Stop
eating my sandwich!” all mean the same thing, but it’s important to be able to know which one to use. Well, if your
sandwich is being eaten, then any of those will probably work.
Math works much the same way. Writing mathematical expressions in different ways is incredibly important,
especially in algebra. It’s not about redundancy; it’s about simplicity. And about keeping a firm grip on that
sandwich of yours.
Students should be able to convert mathematical expressions to alternative but equivalent forms by factoring.
This is important when students want to explain certain properties of an expression or the quantity which the
expression represents, solve equations involving mathematical expressions, or simplify complex expressions. They
might not want to at first, but if you tell them it’s on the chapter test, they’ll most likely be very interested.
Factoring is the conversion of a mathematical expression into an equivalent form that consists of a single term
composed of nothing but terms multiplied together, called factors. Basically, it’s the opposite of the distributive
property. Students should recall that the distributive property is:
a(b + c) = ab + ac
Note that the a has been distributed to the two terms inside the parentheses. Also note that we have taken an
expression with one term and transformed it into an equivalent expression with more than one term. We’ve
expanded the expression.
Factoring is the reverse process, so it can be stated as:
ab + ac = a(b + c)
We have converted from a two-term expression to a one-term expression which consists of nothing but factors,
namely a and (b + c). We repeat the process on any factors which can be factored further. Once we have factored
everything which can be factored, the expression is said to be fully factored. Fancy that.
EXPLANATIONS
AND EXAMPLES
(a.) Example.
Just like English has nouns, verbs, and adjectives, mathematics has terms, factors, and coefficients. Well, sort of.
Students should know that terms are the pieces of the expression that are separated by plus or minus signs,
except when those signs are within grouping symbols like parentheses, brackets, curly braces, or absolute value
bars. Every mathematical expression has at least one term. For instance, the expression 3x + 2 has two terms: 3x
and 2. A term that has no variables is often called a constant because it never changes.
Within each term, there can be two or more factors, the numbers and/or variables multiplied together. The term
3x has two factors: 3 and x. There are always at least two factors, though one of them may be the number 1, which
isn’t usually written. But that 1 is always there…watching us.
Finally, a coefficient is a factor (usually numeric) that is multiplying a variable. Using the example, the 3 in the first
term is the coefficient of the variable x.
The order or degree of a mathematical expression is the largest sum of the exponents of the variables when the
expression is written as a sum of terms. For the example 3x + 2, the order is 1, since the variable x in the first term
has an exponent of 1 and there are no other terms with variables.
The expression 5×2 – 3x + 2 has order 2, whereas the expression 3xy + 5x2y3 – 7x + 32y4 has order 5, because
the exponents of x and y in the second term are 2 and 3, respectively, and 2 + 3 = 5. No other term has a higher
exponent sum.
Now that we have our words, we can start putting them together and make expressions. A good way to see if
students really understand an expression like 3x + 2 is to have them translate mathematical expressions into
English and vice versa. For instance, the expression 3x + 2 could also be written as, “the sum of 3 times a number
and 2,” or, “2 more than three times a number.”
Clearly, it’s much easier to write the mathematical expression than to write it in English (not to mention Pig Latin).
The two are directly related to each other, however, and students should be able to translate back and forth. At
first, students might want to make use of a “dictionary” like the table below to help them go from one language to
the other.
b). Example.
Let’s consider a more complex expression: 5x – (2 – 4y). In English, this could be stated as “the difference between
5 times a number and the quantity ‘4 times another number less than 2’.” That’s a mouthful, and this expression
isn’t even that complex! It should be obvious why we do math in symbol notation now: it’s much easier to write.
Notice how the English expression mentioned “a number” and “another number.” This is a clue that two different
variables must be used in the mathematical expression. These two variables might represent two different physical
quantities in some situation, and the expression shows how each quantity contributes to the overall behavior.
How many terms are in that expression? Your students will probably say three, but there are only two the way the
expression is written. The two terms inside the parentheses are treated as a single thing, so the first term is 5x and
the second term is -(2 – 4y). Since there is no number immediately following the minus sign in the second term,
we assume the number is actually 1. So the second term could be written as -1(2 – 4y). This should be interpreted
as a -1 multiplying everything inside the parentheses.
Of course, we can have expressions, which have even more variables, if there are more changing or unknown
quantities involved. For example, the compound interest expression P(1 + r)n has three variables: P, r, and n, each
representing a different physical quantity. As written, this expression has only one term, consisting of two factors,
P and (1 + r)n. The first factor depends only on P, while the second depends on r and n.
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.SSE.2
Use the structure of an expression to identify ways to rewrite it. For
example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of
squares that can be factored as (x2 – y2)(x2 + y2)
DESCRIPTION
A.SSE.2 Rewrite algebraic expressions in different equivalent forms such as factoring or combining like terms.
A.SSE.2 Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or
difference of two cubes, or a combination of methods to factor completely.
A.SSE.2 Simplify expressions including combining like terms, using the distributive property and other operations
with polynomials.
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 19
MATHEMATICS
ESSENTIAL
QUESTION(S)
• What strategies can be applied to create equivalent expressions?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures. Tasks assessing expressing mathematical reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Identify ways to rewrite
expressions, such as difference
of squares, factoring out a
common monomial, and
regrouping.
Identify various structures of
expressions.
Use the structure of an expression to
identify ways to rewrite it.
Classify expressions by structure
and develop strategies to assist in
classification.
EXPLANATIONS
AND EXAMPLES
Students should extract the greatest common factor (whether a constant, a variable, or a combination of each). If
the remaining expression is quadratic, students should factor the expression further.
In English, we have many different ways of saying the same thing. “Stop that!” and “Stop eating that!” and “Stop
eating my sandwich!” all mean the same thing, but it’s important to be able to know which one to use. Well, if your
sandwich is being eaten, then any of those will probably work.
Math works much the same way. Writing mathematical expressions in different ways is incredibly important,
especially in algebra. It’s not about redundancy; it’s about simplicity. And about keeping a firm grip on that
sandwich of yours.
Students should be able to convert mathematical expressions to alternative but equivalent forms by factoring.
This is important when students want to explain certain properties of an expression or the quantity which the
expression represents, solve equations involving mathematical expressions, or simplify complex expressions. They
might not want to at first, but if you tell them it’s on the chapter test, they’ll most likely be very interested.
Factoring is the conversion of a mathematical expression into an equivalent form that consists of a single term
composed of nothing but terms multiplied together, called factors. Basically, it’s the opposite of the distributive
property. Students should recall that the distributive property is:
a(b + c) = ab + ac
Note that the a has been distributed to the two terms inside the parentheses. Also note that we have taken an
expression with one term and transformed it into an equivalent expression with more than one term. We’ve
expanded the expression.
Factoring is the reverse process, so it can be stated as:
ab + ac = a(b + c)
We have converted from a two-term expression to a one-term expression which consists of nothing but factors,
namely a and (b + c). We repeat the process on any factors which can be factored further. Once we have factored
everything which can be factored, the expression is said to be fully factored. Fancy that.
STANDARD AND DECONSTRUCTION
A.SSE.2
Use the structure of an expression to identify ways to rewrite it. For
example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of
squares that can be factored as (x2 – y2)(x2 + y2)
DESCRIPTION
A.SSE.2 Rewrite algebraic expressions in different equivalent forms such as factoring or combining like terms.
A.SSE.2 Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or
difference of two cubes, or a combination of methods to factor completely.
A.SSE.2 Simplify expressions including combining like terms, using the distributive property and other operations
with polynomials.
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20 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
The factor we pull outside the parenthesis (a, in our previous example) is called the greatest common factor or
GCF if it is the largest factor that is common to all of the terms. The GCF can be a number, a variable, a multi-term
expression, or any combination of these.
The GCF technique is the very first thing that should be checked when confronted with any factoring problem.
Doing so will often simplify subsequent factoring steps. Strong multiplication and division skills are critical to
being efficient at GCF determination, so it may be helpful to go over divisibility rules with your students. (All even
numbers are divisible by 2, if the sum of a number’s digits is divisible by 3 then the number itself is divisible by 3,
and so on.)
Factoring by grouping is a variant of the GCF technique. In this case, we have to group certain terms together,
then treat each grouping as a GCF problem. We continue that process until we just can’t go any further.
For instance, we can factor the expression 8×2 – 4x – 40x + 20 by a common numerical factor of 4. There is no
common variable factor, though. We can factor this as 4(2×2 – x – 10x + 5). What about that mess inside the
parentheses?
If we group the first two terms together, we see they have a common factor of x, so they can be rewritten as x(2x –
1). The last two terms have a common factor of -5. They can then be written as -5(2x – 1). Putting this all together
we get the following:
4[x(2x – 1) – 5(2x – 1)]
Our four-term expression is now a two-term expression. These two terms have a common factor of (2x – 1),
however, so we can factor further:
4(2x – 1)(x – 5)
This is the fully factored form.
As important as GCF factorizations are, students should know how to factor expressions of a higher order. Yes,
we’re talking about exponents.
Students should know that expressions of order two are called quadratic expressions, and that they can be written
in the form ax2 + bx + c, where a, b, and c are numbers and a ≠ 0. (Ask the students to explain why that restriction
is necessary.)
Factoring quadratics is a little more challenging, so it helps to examine the factoring process in reverse. Consider
the expression (2x – 5)(3x + 2). Clearly, this is a factored form of a quadratic expression. If we use the distributive
property to expand the expression we would get
(2x – 5)(3x + 2)
2x(3x + 2) – 5(3x + 2)
6×2 + 4x – 15x – 10
6×2 – 11x – 10
The last expression is the quadratic expression written in standard form. We can identify a = 6, b = -11, and c = -10.
Notice that if we run this distributive process in reverse, it’s like taking the standard form quadratic and splitting
up the middle term into two pieces and then using the grouping technique. How would we know to break up the
middle term in that way, though? There are many ways to write -11 as the sum of two numbers, but only one will
give us what we need.
How would we know to pick +4 and -15? Notice that the product of +4 and -15 is -60. This is also the product of
the first and last coefficients in the standard form, a and c: (6)(-10) = -60. So, one might assume we multiplied a
and c together, then found two numbers which multiply together to get that number but add together to give b. If
you study several examples, you’ll see this is correct. This is always the case. So the steps in the general technique
for factoring quadratics which can be factored are:
1. Write the quadratic in standard form.
2. Multiply a and c together.
3. Find two numbers which multiply to produce the number in step 2, but add to produce b. Be careful with
signs!
4. Rewrite the quadratic, but replace the middle term with two terms using the two numbers from step 3.
5. Use the grouping technique to factor the expression.
Students should know that this general technique works for all factorable quadratics, but that there are some
special cases to consider. For instance, the difference of two squares x2 – y2 can be factored as (x – y)(x + y). Also, the
perfect trinomial x2 + 2ax + a2 can be factored to equal (x + a)2. These two work even when x, y, and a are quantities
made of multiple terms. That’s pretty nifty.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 21
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
In general, students should follow these rules:
1. Determine if there is a greatest common factor in the expression. You typically want to factor that out first,
as it leaves a simpler expression to concentrate on in subsequent steps.
2. Determine if it is possible to use the grouping technique.
3. If the expression is a quadratic, put it in standard form first.
4. Examine the form of the expression and determine if one of the special forms—the difference of two
squares or the perfect square trinomial—applies. If so, apply the rules given previously for writing the
factored form.
5. If the quadratic is not a special form, use the general method for factoring quadratics.
Factoring takes a lot of practice, but it is probably the single most important skill students will develop in math.
It is critical to just about every topic in higher levels of math, so it is worthwhile to spend a fair amount of time
developing this skill. Once students start getting the hang of it, many of them will be able to just “see” the factors.
Encourage them to always look for patterns, to pay attention to the individual pieces of the expressions they get
and how they appear to relate to the pieces of the original expression.
(Source: www.shmoop.com)
Example:
Factor x3 – 2×2 – 35x
In the traditional pathway, linear, quadratic, and exponential expressions are the focus in Algebra I, and integer
exponents are extended to rational exponents (only those with square or cubed roots). In Algebra II, the
expectation is to extend to polynomial and rational expressions.
In the international pathway, CCSS Mathematics I focuses on linear expressions and exponential expressions
with integer exponents. CCSS Mathematics II extends exponential expressions from the first course and includes
quadratic expressions. Course II also extends integer exponents to rational exponents, focusing on those with
square or cubed roots. In CCSS Mathematics III the expectation is polynomial and rational expressions.
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http://www.shmoop.com
22 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
CLUSTER 2. Write expressions in equivalent forms to solve problems.
BIG IDEA • Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
STANDARD AND DECONSTRUCTION
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression. («)
DESCRIPTION
A.SSE.3a Write expressions in equivalent forms by factoring to find the zeros of a quadratic function and explain the
meaning of the zeros.
Given a quadratic function explain the meaning of the zeros of the function. That is if f(x) = (x – c) (x – a) then f(a) = 0
and f(c) = 0.
Given a quadratic expression, explain the meaning of the zeros graphically. That is for an expression (x – a) (x – c), a
and c correspond to the x-intercepts (if a and c are real).
A.SSE.3b Write expressions in equivalent forms by completing the square to convey the vertex form, to find the
maximum or minimum value of a quadratic function, and to explain the meaning of the vertex.
A.SSE.3c Use properties of exponents (such as power of a power, product of powers, power of a product, and
rational exponents, etc.) to write an equivalent form of an exponential function to reveal and explain specific
information about its approximate rate of growth or decay.
ESSENTIAL
QUESTION(S)
• What strategies can be applied to create equivalent expressions that lead to accuracy and efficiency?
• What strategies can be used to factor a quadratic equation?
• What does completing the square in a quadratic expression reveal?
• What is the most efficient way to transform expressions for exponential functions?
MATHEMATICAL
PRACTICE(S)
HS.MP.1. Make sense of problems and persevere in solving them.
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
SUBSTANDARD
DECONSTRUCTED
A.SSE.3a. Factor a quadratic expression to reveal the zeros of the function
it defines.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Explain the connection between
the factored form of a quadratic
expression and the zeros of the
function it defines.
Explain the properties of the
quantity represented by the
quadratic expression.
Factor a quadratic expression to
produce an equivalent form of the
original expression.
Choose and produce an equivalent
form of a quadratic expression to
reveal and explain properties of the
quantity represented by the original
expression.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 23
MATHEMATICS
STANDARD AND DECONSTRUCTION
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression. («)
DESCRIPTION
A.SSE.3a Write expressions in equivalent forms by factoring to find the zeros of a quadratic function and explain the
meaning of the zeros.
Given a quadratic function explain the meaning of the zeros of the function. That is if f(x) = (x – c) (x – a) then f(a) = 0
and f(c) = 0.
Given a quadratic expression, explain the meaning of the zeros graphically. That is for an expression (x – a) (x – c), a
and c correspond to the x-intercepts (if a and c are real).
A.SSE.3b Write expressions in equivalent forms by completing the square to convey the vertex form, to find the
maximum or minimum value of a quadratic function, and to explain the meaning of the vertex.
A.SSE.3c Use properties of exponents (such as power of a power, product of powers, power of a product, and
rational exponents, etc.) to write an equivalent form of an exponential function to reveal and explain specific
information about its approximate rate of growth or decay.
ESSENTIAL
QUESTION(S)
• What strategies can be applied to create equivalent expressions that lead to accuracy and efficiency?
• What strategies can be used to factor a quadratic equation?
• What does completing the square in a quadratic expression reveal?
• What is the most efficient way to transform expressions for exponential functions?
MATHEMATICAL
PRACTICE(S)
HS.MP.1. Make sense of problems and persevere in solving them.
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
SUBSTANDARD
DECONSTRUCTED
A.SSE.3a. Factor a quadratic expression to reveal the zeros of the function
it defines.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Explain the connection between
the factored form of a quadratic
expression and the zeros of the
function it defines.
Explain the properties of the
quantity represented by the
quadratic expression.
Factor a quadratic expression to
produce an equivalent form of the
original expression.
Choose and produce an equivalent
form of a quadratic expression to
reveal and explain properties of the
quantity represented by the original
expression.
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24 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
SUBSTANDARD
DECONSTRUCTED
A.SSE.3b. Complete the square in a quadratic expression to reveal the
maximum or minimum value of the function it defines.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Explain the connection between
the completed square form of
a quadratic expression and the
maximum or minimum value of the
function it defines.
Complete the square on a
quadratic expression to produce an
equivalent form of an expression.
Explain the properties of the
quantity or quantities represented
by the transformed exponential
expression.
Choose and produce an equivalent
form of an exponential expression
to reveal and explain properties of
the quantity represented by the
original expression.
SUBSTANDARD
DECONSTRUCTED
A.SSE.3c. Use the properties of exponents to transform expressions for
exponential functions. For example the expression 1.15t can be rewritten
as (1.151⁄12)12t ≈1.01212t to reveal the approximate equivalent monthly interest
rate if the annual rate is 15%.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Use the properties of exponents to
transform simple expressions for
exponential functions.
Choose and produce an equivalent
form of a quadratic expression to
reveal and explain properties of the
quantity represented by the original
expression.
EXPLANATIONS
AND EXAMPLES
Students will use the properties of operations to create equivalent expressions.
Your students should be gaining fluency in mathematics, and be able to write and rewrite expressions. The spells
of nausea, hyperventilation, and paranoia surrounding the beautiful language of mathematics should have
subsided by now. If they haven’t, you may want to consult the school nurse.
Rather than rewriting expressions for the fun of it (and it is fun, isn’t it?), students should understand what these
different expressions can tell us about the quantities they represent. These mathematical expressions can tell us
the zeros (or roots or x-intercepts) and the maximum and minimum values of a function, and have plenty of other
applications in the real world. Most of them involving money.
(a.) Example.
Students should already be comfortable with finding the zeros of functions. In other words, finding the x values
that make the mathematical expression equal to 0. For linear expressions like 5x – 10, it’s relatively easy to find the
zeros. All we have to do is set the expression to equal 0 and solve for x. In this case, x = 2. Duh.
But what about quadratic expressions that aren’t that simple? The simplest way to find roots for higher order
expressions is by factoring. We can actually demonstrate that using our linear expression. Recall that we set our
linear expression equal to 0 in order to find the root. Mathematically, we had 5x – 10 = 0. Your students most likely
added 10 to both sides and then divided by 5, but let’s give factoring a shot.
We can factor that expression by pulling out 5 (the coefficient of x) as the GCF. That would yield the factored form
5(x – 2) = 0. We’ve turned our two-term expression into a one-term expression which has two factors, 5 and (x – 2).
The product of those factors is zero, however. By the zero product rule, at least one of the factors must be zero.
Well, obviously, 5 can never be 0. Therefore, x – 2 must be equal to zero. If we solve the equation x – 2 = 0, we get x
= 2, which is the answer we got before.
So how does this work for a quadratic? When finding the zeros of a quadratic equation, the best thing to do is try
and factor. We can take an expression like 6×2 – 11x – 10 and turn it into (2x – 5)(3x + 2). To find the zeros of the
expression, we set this factored form equal to zero, then apply the zero product rule.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 25
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
So we have to solve two first order equations: 2x – 5 = 0 and 3x + 2 = 0. The first one gives. The second one gives
. There should be two roots, since this is a second order expression, and these are the two roots we are looking for.
The equation y = 6×2 – 11x – 10 has two x-intercepts! The graph of that equation looks like this:
It crosses the x-axis two times, and at the locations we calculated. This curve is called a parabola. All quadratic
expressions of this type have the same basic shape.
Sometimes, quadratics have only one x-intercept and other times they don’t have any real roots at all. It just means
the function doesn’t cross the x-axis, or that it crosses the x-axis once (at the minimum or maximum). Make sure
your students don’t have a panic attack when that happens.
(b) Example.
Before you talk about completing the square, remind your students about the perfect square trinomial. In general
form, a perfect square trinomial is a quadratic of the form m2 ± 2mn + n2, which can be factored into the form (m ±
n)(m ± n) = (m ± n)2.
Notice that half of the middle term is equal to the square root of the first term times the square root of the last
term. This is a requirement for a quadratic to be a perfect square trinomial. Not all quadratics are in this form.
Some algebraic manipulation can be used to put any quadratic equation into a form where one side is a perfect
square trinomial. That’s called completing the square.
What about the equation y = x2 – 4x + 3? This quadratic is factorable as (x – 1)(x – 3). Clearly not a perfect square
trinomial, but no one’s perfect, right?
First, subtract 3 from both sides to get y – 3 = x2 – 4x. Now, we can add whatever we want to the right-hand side,
as long as we do exactly the same thing to the left-hand side. We want a 4 where the 3 used to be (because 4 is
a perfect square of 2), so we get y – 3 + 4 = x2 – 4x + 4. The right-hand side is now in the proper perfect square
trinomial form, and can be factored as (x – 2)2. Our equation is then y + 1 = (x – 2)2.
For every x-value we plug into the right hand side of the equation, there is a corresponding y-value on the left-
hand side. When x = -2, y must be equal to 15, for example. Notice that the right hand side of the equation is never
negative, since we are squaring a number, and that always produces a positive number.
The smallest thing the right-hand side can possibly be is 0 (when x = 2). This is going to translate to the smallest
possible value that y can be (y + 1 = 0, which means y = -1). In mathematical terms, that point represents the
minimum of the function. In this case, the point (2, -1) represents the minimum point of our parabola, which is
called the vertex.
Every quadratic equation has either a minimum or a maximum, depending on whether it points up or down
(whether it looks like a happy face or a sad face). Notice that we could not have easily found the coordinates of the
vertex from y = (x – 1)(x – 3). We had to put the quadratic in a different form that highlighted the coordinates of
the vertex in order to extract this information.
(c) Example.
At this point, the students should have a firm understanding of the concept of an exponent—a power to which
some other quantity is raised. Up until this point, however, exponents probably have always been numbers.
Students may not be aware that exponents can be variables, too.
Consider an expression like rn. The r is identified as the base of the expression, while the n is identified as the
exponent. Since both r and n are variables, this is a more general type of expression involving exponents than
something like r2 or x10, where the base is the variable and the exponent is a number. These general expressions
are known as (surprise, surprise) exponential expressions, while the others are usually referred to as power law
expressions.
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26 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
Students should know that the rules for manipulating power law expressions apply for exponential expressions
too. The following table summarizes those rules:
In multiplying and dividing exponential expressions, just as in power law expressions, the bases have to be the
same. We can use the rules for exponential expressions to transform them into alternate forms, allowing us to
interpret the expressions in different ways.
Everyone loves earning money, so let’s find out ways to do that. No, we don’t mean playing the lotto, because
that’s all chance and probability. We want exponential growth, so we can earn some dinero the good old-
fashioned way: investing.
The compound interest expression allows us to calculate the new value of an investment after a certain period of
time has elapsed if interest is earned at a certain rate for each period of time. It sounds really complicated, but it
looks pretty simple: P(1 + r)n. In this expression, P is the initial amount or present value, r is the interest rate for each
period expressed in decimal form, and n is the number of periods over which the new value is to be calculated.
Let’s focus on the second factor of that expression: (1 + r)n. As an example, let’s say we open a savings account,
which offers a rate of 15% per year. In other words, our money grows by 15% each year. This is represented by the
mathematical expression (1 + 0.15)n = (1.15)n, where n is the number of years over which we allow this account to
grow.
What if we were thinking about switching banks, and they offered a savings account, but expressed their rates on
a per month basis rather than a per year basis? Obviously, we want to get the highest rate we can, since that’s how
we make more money. How could we compare the two?
We use the laws of exponentials. If we want our periods to be in months, we have to transform our exponential
expression to be on a per month basis. There are 12 months in a year. So, if the account accrues interest for n years,
it accrues interest for 12n months. The exponent of our transformed expression must be 12n.
We can’t just change the exponent without also changing the part inside the parentheses, though, because we
would be creating a completely different expression. To balance multiplying the exponent by 12, we have to
divide any exponent inside the parentheses by 12. Even though it isn’t written, there is an exponent there, since
1.151 = 1.15. So, the new exponent inside the parentheses must become , and the transformed exponential
expression is .
Using the rule that when raising an exponential expression to a power you multiply the exponents, the students
can verify that we get the original expression back. We use a calculator to evaluate . The final
expression, in terms of monthly periods is (1.012)12n = (1 + 0.012)12n. If the annual rate is 15%, the equivalent
monthly rate is approximately 1.2%, and we can now compare this to what the other bank offers.
(Source: www.shmoop.com)
ESSENTIAL
QUESTION(S)
• Why can rewriting exponential expressions in another form lead to efficiency when solving a problem?
MATHEMATICAL
PRACTICE(S)
HS.MP.3. Construct viable arguments and critique the reasoning of others.
HS.MP.4. Model with mathematics.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Find the first term in a geometric
sequence given at least two other
terms..
Define a geometric series as a series
with a constant ratio between
successive terms.
Use the formula S + a (1-rn)/(1-r) to
solve problems.
Derive a formula [i.e. equivalent to
the formula S + a (1-rn)/(1-r) ] for
the sum of a finite geometric series
(when the common ratio is not 1).
EXPLANATIONS
AND EXAMPLES
Although students may not know it, they already know a lot about sequences and series. Not only are sequences
and series on TV, in music, and all over the Internet, they were also taught between naptime and making those
dried macaroni picture frames. You could have a sequence as easy as one, two, three.
In mathematics, a sequence is a bunch of numbers listed one after the other. In many cases, it is possible to
determine a particular member of a sequence simply from its location, just like when counting natural numbers.
That sequence starts with the number 1 and every other member of the sequence can be found by taking the
previous member of the sequence and adding 1 to it.
Students should know the difference between an arithmetic sequence and a geometric sequence. In arithmetic
sequences, successive members have a common difference. That is, the difference between each member and the
one before it is some constant value. In geometric sequences, all numbers have a common ratio, meaning that the
quotient of each member and the one before it is some constant value.
It’s best to give several examples of both arithmetic and geometric sequences and series, and then finally give a
formula for each. For instance, if a is the first term in an arithmetic series and the common difference is d, then the
series would be the sum of a, a + d, a + 2d, a + 3d, and so on. For a geometric sequence with common ratio r, we’ll
have a, ar, ar2, ar3, ar4, ar5, and so on.
Notice that the first member of the sequence has no factors of r. The second member has 1 factor of r, the third has
2 factors of r, the fourth has 3 factors of r, and so on. The number of factors of r is always one less than the number
of that particular term in the sequence. If n represents the number of the nth member of the sequence, then that
number has a value of arn – 1.
Students should know that a series is the sum of the members of a sequence. We can form sums of both arithmetic
and geometric sequences. For an n-term geometric sequence, we can write the series as sn = a + ar + ar2 + ar3 + ar4
+ ar5 + … + arn – 1.
STANDARD AND DECONSTRUCTION
A.SSE.4
Derive the formula for the sum of a finite geometric series (when the
common ratio is not 1), and use the formula to solve problems. For
example, calculate mortgage payments. («)
DESCRIPTION
A.SSE.4 Develop the formula for the sum of a finite geometric series when the ratio is not 1.
A.SSE.4 Use the formula to solve real world problems such as calculating the height of a tree after n years given the
initial height of the tree and the rate the tree grows each year. Calculate mortgage payments.
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 27
MATHEMATICS
ESSENTIAL
QUESTION(S)
• Why can rewriting exponential expressions in another form lead to efficiency when solving a problem?
MATHEMATICAL
PRACTICE(S)
HS.MP.3. Construct viable arguments and critique the reasoning of others.
HS.MP.4. Model with mathematics.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Find the first term in a geometric
sequence given at least two other
terms..
Define a geometric series as a series
with a constant ratio between
successive terms.
Use the formula S + a (1-rn)/(1-r) to
solve problems.
Derive a formula [i.e. equivalent to
the formula S + a (1-rn)/(1-r) ] for
the sum of a finite geometric series
(when the common ratio is not 1).
EXPLANATIONS
AND EXAMPLES
Although students may not know it, they already know a lot about sequences and series. Not only are sequences
and series on TV, in music, and all over the Internet, they were also taught between naptime and making those
dried macaroni picture frames. You could have a sequence as easy as one, two, three.
In mathematics, a sequence is a bunch of numbers listed one after the other. In many cases, it is possible to
determine a particular member of a sequence simply from its location, just like when counting natural numbers.
That sequence starts with the number 1 and every other member of the sequence can be found by taking the
previous member of the sequence and adding 1 to it.
Students should know the difference between an arithmetic sequence and a geometric sequence. In arithmetic
sequences, successive members have a common difference. That is, the difference between each member and the
one before it is some constant value. In geometric sequences, all numbers have a common ratio, meaning that the
quotient of each member and the one before it is some constant value.
It’s best to give several examples of both arithmetic and geometric sequences and series, and then finally give a
formula for each. For instance, if a is the first term in an arithmetic series and the common difference is d, then the
series would be the sum of a, a + d, a + 2d, a + 3d, and so on. For a geometric sequence with common ratio r, we’ll
have a, ar, ar2, ar3, ar4, ar5, and so on.
Notice that the first member of the sequence has no factors of r. The second member has 1 factor of r, the third has
2 factors of r, the fourth has 3 factors of r, and so on. The number of factors of r is always one less than the number
of that particular term in the sequence. If n represents the number of the nth member of the sequence, then that
number has a value of arn – 1.
Students should know that a series is the sum of the members of a sequence. We can form sums of both arithmetic
and geometric sequences. For an n-term geometric sequence, we can write the series as sn = a + ar + ar2 + ar3 + ar4
+ ar5 + … + arn – 1.
STANDARD AND DECONSTRUCTION
A.SSE.4
Derive the formula for the sum of a finite geometric series (when the
common ratio is not 1), and use the formula to solve problems. For
example, calculate mortgage payments. («)
DESCRIPTION
A.SSE.4 Develop the formula for the sum of a finite geometric series when the ratio is not 1.
A.SSE.4 Use the formula to solve real world problems such as calculating the height of a tree after n years given the
initial height of the tree and the rate the tree grows each year. Calculate mortgage payments.
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28 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
This is all fine if we have 6 or 7 terms, but imagine trying to use this to calculate a series with 100 terms. Or 200
terms. Or an infinite number of terms. That would be an awful, not to mention time-consuming calculation to do
by hand. Luckily, we don’t have to, because we can perform a little magic to greatly simplify this expression. (It’s
actually not magic, but don’t tell your students that. Some of them are still waiting for their invitation to Hogwarts,
and this is the closest they’ll ever get.)
Let’s take the expression for the series and multiply it by the common ratio, r.
rsn = ar + ar2 + ar3 + ar4 + ar5 + ar6 + … + arn
Notice that the number of factors of r in each term in the sum increased by 1. Now, let’s take our original
expression for s and subtract this new expression from it. There is nothing that says we can’t do that, right?
sn – rsn = (a + ar + ar2 + ar3 + ar4 + ar5 + … + arn – 1) – (ar + ar2 + ar3 + ar4 + ar5 + ar6 + … + arn)
Notice that except for the very first term in the expression for s and the very last term in the expression for rs, there
are matching terms in the two sets of parentheses. That means all of those terms cancel each other out, and we’re
left with sn – rsn = a – arn.
We can then factor the left-hand side of that equation as sn(1 – r). Dividing both sides by 1 – r and factoring out
the a in the numerator on the right side gives us a much simpler way to find the sum of a geometric series.
Your students should be applauding right about now. We know it’s tough to compete with Hogwarts, but come
on. This math is clearly not meant for mere muggles.
We do have to be careful about one thing, though. Notice that we divide by 1 – r. If r = 1, we have a serious
problem since that’s dividing by 0. So we have to place a restriction on this formula, since using it when r = 1
might bring back He-Who-Must-Not-Be-Named…again.
Geometric series appear in many places, but one case which touches just about everyone is in the area of finance
and economics. Calculations involving loan payments (for a mortgage, for instance) often make use of geometric
series.
Suppose you took out a $200,000 loan for a dragon. It sounds expensive, but it’s worth it if you’re a frequent flyer.
Plus, it’s a dragon.
The beastly $200,000 loan has a 30-year fixed interest rate of 3.6% per year paid monthly. By convention, this
means that each month, an interest rate of 0.036 ÷ 12 = 0.003, or 0.3% will be applied to the unpaid principal to
calculate the interest payment for that month. This will continue for each month until the term of 30 years has
been reached, which is (12)(30) = 360 months. If m is the monthly amount, P is the principal amount on the loan, r
is the rate, and n is the number of months, we can use the following formula to find the monthly amount.
There are calculators online, which will do this sort of calculation for you, but it is nice to understand how the
calculation is done so your students can verify it for themselves. As far as dragons go, nobody wants to pay more
than they have to. On second thought, that probably applies to just about anything.
(Source: www.shmoop.com)
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 29
MATHEMATICS
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MATHEMATICS
ARITHMETIC
WITH POLYNOMIALS
AND RATIONAL
EXPRESSIONS
(A-APR)
DOMAIN:
HIGH SCHOOL
ALGEBRA
32 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
DOMAIN Arithmetic with Polynomials and Rational Expressions (A-APR)
CLUSTERS
1. Perform arithmetic operations on polynomials.
2. Understand the relationship between zeros and factors of polynomials.
3. Use polynomial identities to solve problems.
4. Rewrite rational expressions.
ACADEMIC
VOCABULARY
Algebra I
absolute value, complement of an event, compound, conjunction, direct and inverse variation, disjunction,
domain & range, exponential growth (and decay), interest (simple and compound), irrational numbers, joint and
conditional probability, law of large numbers, mathematical model, measure of spread (range, interquartile range),
midpoint formula, outlier, parent function, Pascal’s triangle, polynomial (binomial, trinomial), quadratic formula
(including discriminant), quantitative and qualitative data, radicand, rational expression, real number properties,
real roots (zeros, solutions, x-intercepts), relative frequency, sequences (arithmetic, geometric, Fibonacci),
simulations, subsets of real numbers
Algebra II
amplitude, asymptote, binomial theorem, combination, common ratio (geometric sequence), complete the
square, complex conjugate, complex number, composition (of functions), conic sections (circles, parabola, ellipse,
hyperbola), empirical rule, factorial, focus (pl. foci), independent and dependent events, inverse of a relation,
logarithm, normal distribution, period, permutation, piece-wise function, radian measure, rational function,
regression equation, series (arithmetic, geometric, finite, infinite, etc.), sigma, standard deviation, step function,
synthetic division, transcendental function, trigonometric function, trigonometric identity, unit circle, variance
CLUSTER 1. Perform arithmetic operations on polynomials
BIG IDEA
• The operations of addition, subtraction and multiplication can be applied to polynomials.
• Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
ESSENTIAL
QUESTION(S)
• What strategies can be used to perform arithmetic operations on polynomials?
MATHEMATICAL
PRACTICE(S)
HS.MP.8. Look for regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures. Tasks assessing expressing mathematical reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Define closure.
Identify that the sum,
difference, or product of two
polynomials will always be a
polynomial, which means that
polynomials are closed under
the operations of addition,
subtraction, and multiplication.
Apply arithmetic operations
of addition, subtraction, and
multiplication to polynomials.
EXPLANATIONS
AND EXAMPLES
Students should understand that polynomials, like integers, are “closed” when it comes to addition, subtraction,
and multiplication. Basically, this just means they’re kind of cliquey as far as these operations are concerned.
An integer plus an integer is an integer, an integer minus an integer is an integer, and an integer times an integer
is an integer. Similarly, a polynomial plus a polynomial is a polynomial, a polynomial minus a polynomial is a
polynomial, and a polynomial times a polynomial is a polynomial. If that isn’t cliquey, we don’t know what is.
Students should know that a polynomial is any expression that is a combination of more than one term via
addition or subtraction. Each individual term is called a monomial. Monomials can be constant (like single
numbers) or include variables to different degrees (like x6). As long as it’s in one lump with no plus or minus signs,
it’s a monomial.
Some examples of polynomials include:
• x + 4
• x2 + 2
• 2×2 – 3x + 5
• x6 + 4×5 – 3×2 + x
STANDARD AND DECONSTRUCTION
A.APR.1
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.
DESCRIPTION
A.APR.1 Understand the definition of a polynomial.
A.APR.1 Understand the concepts of combining like terms and closure.
A.APR.1 Add, subtract, and multiply polynomials and understand how closure applies under these operations.
Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.
In the traditional pathway, linear and quadratic polynomial expressions are the expectation in Algebra I and beyond
quadratic polynomial expressions is the expectation for Algebra II.
In the international pathway, CCSS Mathematics II places emphasis on polynomials that simplify to quadratics and
CCSS Mathematics III extends beyond quadratics.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 33
MATHEMATICS
ESSENTIAL
QUESTION(S)
• What strategies can be used to perform arithmetic operations on polynomials?
MATHEMATICAL
PRACTICE(S)
HS.MP.8. Look for regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures. Tasks assessing expressing mathematical reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Define closure.
Identify that the sum,
difference, or product of two
polynomials will always be a
polynomial, which means that
polynomials are closed under
the operations of addition,
subtraction, and multiplication.
Apply arithmetic operations
of addition, subtraction, and
multiplication to polynomials.
EXPLANATIONS
AND EXAMPLES
Students should understand that polynomials, like integers, are “closed” when it comes to addition, subtraction,
and multiplication. Basically, this just means they’re kind of cliquey as far as these operations are concerned.
An integer plus an integer is an integer, an integer minus an integer is an integer, and an integer times an integer
is an integer. Similarly, a polynomial plus a polynomial is a polynomial, a polynomial minus a polynomial is a
polynomial, and a polynomial times a polynomial is a polynomial. If that isn’t cliquey, we don’t know what is.
Students should know that a polynomial is any expression that is a combination of more than one term via
addition or subtraction. Each individual term is called a monomial. Monomials can be constant (like single
numbers) or include variables to different degrees (like x6). As long as it’s in one lump with no plus or minus signs,
it’s a monomial.
Some examples of polynomials include:
• x + 4
• x2 + 2
• 2×2 – 3x + 5
• x6 + 4×5 – 3×2 + x
STANDARD AND DECONSTRUCTION
A.APR.1
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.
DESCRIPTION
A.APR.1 Understand the definition of a polynomial.
A.APR.1 Understand the concepts of combining like terms and closure.
A.APR.1 Add, subtract, and multiply polynomials and understand how closure applies under these operations.
Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.
In the traditional pathway, linear and quadratic polynomial expressions are the expectation in Algebra I and beyond
quadratic polynomial expressions is the expectation for Algebra II.
In the international pathway, CCSS Mathematics II places emphasis on polynomials that simplify to quadratics and
CCSS Mathematics III extends beyond quadratics.
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34 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
Polynomials are really nice to work with because they’re continuous and defined for all values. In other words, we
can replace x with any real number, and we’ll get a real number as our result. Just don’t rub it in Pinocchio’s ever-
growing nose. He’s always wanted to be real.
For example, take the polynomial x3 + 2x – 5. Input any value of x, like 6, and we’ll get a real number, like (6)3 + 2(6)
– 5 = 223. Students should know that adding, subtracting, and multiplying two or more polynomials together will
give them a polynomial. A different polynomial, but still a polynomial.
However, polynomials are not closed (so they’re…open?) under division because sometimes the quotient won’t
be another polynomial. Take this quotient of polynomials, for example.
This is a rational expression, not a polynomial. Somehow, polynomials seem a lot more rational.
Definitions are crucial for students to understand before learning how to actually perform operations on
polynomials.
Students should know that when adding and subtracting polynomials, we can only combine like terms with
like terms. Constants can only be added to constants, x terms can only be added to x terms, x2 terms can only be
added to x2 terms, and so on.
If addition and subtraction are like OCD, meticulously pairing terms that go together, then multiplication is like
ADHD, combining any and all terms together in one big dog pile regardless of what they are.
Students should know that multiplying a polynomial by a monomial means distribution, and that multiplying two
polynomials together means a lot of distribution. More specifically, we have to make sure to multiply every term in
one polynomial by every term in the other polynomial.
For instance, students performing the operation (x + 2) × (x3 + x – 7) should know to first distribute the x to get x4
+ x2 – 7x, and then the 2 to get 2×3 + 2x – 14, and then to add the two together so that the final answer is x4 + 2×3 +
x2 – 5x – 14.
When two binomials are multiplied together, like (x + 1)(x + 3), most students prefer to remember the acronym
FOIL, which stands for multiplying the First, Outer, Inner, and Last numbers together.
The best way to get students to feel comfortable is to have them practice, but have them stick to operations like
addition, subtraction, and multiplication at first. They’ll get to more complicated operations such as lobotomies
and appendectomies later…like, after med school..
(Source: www.shmoop.com)
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 35
MATHEMATICS
CLUSTER 2. Understand the relationship between zeros and factors of polynomials
BIG IDEA
• The operations of addition, subtraction and multiplication can be applied to polynomials.
• Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
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36 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How are zeros and factors of a polynomial related?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.3. Construct viable arguments and critique the reasoning of others.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures. Tasks assessing expressing mathematical reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Define the remainder theorem
for polynomial division and
divide polynomials.
Given a polynomial p(x) and
a number a, divide p(x) by
(x – a) to find p(a) then apply
the remainder theorem and
conclude that p(x) is divisible
by x – a, if and only if p(a) = 0.
EXPLANATIONS
AND EXAMPLES
Before introducing remainders of polynomials to your students, be sure they remember what a remainder is. No,
not a reindeer. A remainder.
For example, tell them to divide 13 by 4. If they’ve gotten this far in math, this really shouldn’t be a problem for
them. Most of them will give you the answer 3.25. Instead, backtrack a few years of schooling and tell them to
write their answer as “3 remainder 1.” It’s the same thing, only for third graders.
If we divide two integers, sometimes they make another integer (6 ÷ 2 = 3), and other times they have remainders
(13 ÷ 4 = 3 remainder 1). A remainder of 0 means that the second number is a factor of first number. For instance, 2
is a factor of 6 because we can multiply 2 by an integer to get 6.
Once they’ve gotten over the intense déjà-vu, slowly and gently explain to them that polynomials are the same
way. If dividing polynomial p(x) by x – a has a remainder of 0, we’ll know that x – a is a factor of p(x). In other words,
p(x) = q(x) × (x – a) where q(x) is a polynomial or an integer.
A remainder of 0 also means that if we substitute a for x, we’ll end up with p(a) = 0 regardless of what q(a) is. That’s
because p(a) = q(a) × (a – a) = q(a) × 0. Hopefully, by now your students know that anything multiplied by zero is
zero.
If p(x) divided by x – a has a remainder, we can write the equation as p(x) = q(x) × (x – a) + r(x), where r(x) is the
remainder. Sometimes r(x) is a number and other times it’s another polynomial. If we set x = a in this situation, we’ll
end up with p(a) = r(a). Would you look at that?
Essentially, any polynomial p(x) can be written as a product of (x – a) and some quotient q(x), plus the remainder
p(a).
p(x) = q(x) × (x – a) + p(a)
When p(a) = 0, (x – a) is a factor of p(x).
STANDARD AND DECONSTRUCTION
A.APR.2
Know and apply the Remainder Theorem: For a polynomial p(x) and a
number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if
(x – a) is a factor of p(x).
DESCRIPTION
A.APR.2 Understand and apply the Remainder Theorem.
A.APR.2 Understand how this standard relates to A.SSE.3a.
A.APR.2 Understand that a is a root of a polynomial function if and only if x-a is a factor of the function.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 37
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
Students should know how to perform all these calculations and rearrangements. For their answers to be at least
somewhat reasonable, they should be comfortable with factoring polynomials, finding remainders, and of course,
division. You’d think that last part goes without saying, but you never know.
If students find these tasks particularly confusing, we suggest practicing with integers. That way, students can
perform these same calculations within their comfort zones. Plus, they can check their answers, too. For instance,
8⁄6 = 1 remainder 2
Converting this into the formula, we get 8 = 1(6) + 2 because 8 is like p(x), 1 is like q(x), and 2 is like p(a). Make
sense? Now get them to try some polynomials instead.
(Source: www.shmoop.com)
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38 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How can a graph of a function be estimated based on the zeros and factors of a polynomial?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Factor polynomials using any available
method.
Create a sign chart for a polynomial f(x)
using the polynomial’s x-intercepts and
testing the domain intervals for which f(x)
greater than and less than zero.
Use the x-intercepts of a polynomial
function and the sign chart to construct a
rough graph of the function.
EXPLANATIONS
AND EXAMPLES
Graphing calculators or programs can be used to generate graphs of polynomial functions.
In plain English, this standard is all about factoring polynomials and finding their zeros. It’s about time too, because it
can be frustrating dealing with students who claim that x3 – 10×2 – 2x + 24 has 1 zero because of 10×2. So frustrating
that you may or may not want to quit this teaching gig and get a job at the local circus. At least they’d take you more
seriously there.
First, your students should know that finding the zeros of a polynomial doesn’t mean counting how many zeros they
can find. That’s not algebra. That’s counting.
The zeros of a polynomial are the x values when we set the polynomial itself to equal zero. In other words, when we
plug in any of the zeros of a polynomial in for x, our answer should be 0. So the zeros of x3 – 10×2 – 2x + 24 are the x
values that make the equation x3 – 10×2 – 2x + 24 = 0 true.
Why are these zero values important? On the coordinate plane, they’re the places where the function crosses the
x-axis. The zeros of the polynomial (also called the solutions or “roots”) are the x-intercepts of the graph.
To find the zeros, students should factor polynomials into linear factors (where x is only to the first power). Then,
setting each linear factor equal to 0 and solving for the variable will give us the x-intercepts, which can be used to
graph the function.
Students should also have a rough idea of how different polynomials should look when graphed. Linear functions
make lines, quadratic functions make parabolas, and so on. They don’t have to know exactly which term affects
which aspect of the graph, but their answers shouldn’t look like a game of connect-the-dots.
For example, we’re told to graph the function defined by the polynomial x2 + 5x + 6. Students should feel
comfortable factoring this into a product of two linear factors: (x + 2)(x + 3). Solving the equation x2 + 5x + 6 = 0
would be difficult, but (x + 2)(x + 3) = 0 is much easier.
The entire polynomial equals 0 when each individual factor equals 0. That gives us x + 2 = 0 and x + 3 = 0, or x = -2
and x = -3. So our x-intercepts are at x = -2 and x = -3. Knowing these intercepts makes it easier to graph the function.
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.APR.3
Identify zeros of polynomials when suitable factorizations are available,
and use the zeros to construct a rough graph of the function defined by the
polynomial.
DESCRIPTION
A.APR.3 Find the zeros of a polynomial when the polynomial is factored.
A.APR.3 Use the zeros of a function to sketch a graph of the function.
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 39
MATHEMATICS
CLUSTER 3. Use polynomial identities to solve problems
BIG IDEA
• The operations of addition, subtraction and multiplication can be applied to polynomials.
• Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
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40 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How can proving polynomial identities help solve problems efficiently?
MATHEMATICAL
PRACTICE(S)
HS.MP.7. Look for and make use of structure.
HS.MP.8. Look for and express regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Explain that an identity shows a
relationship between two quantities or
expressions, which is true for all values of
the variables, over a specified set.
Prove polynomial identities.
Use polynomial identities to
describe numerical relationships.
EXPLANATIONS
AND EXAMPLES
With the increase in technology and this huge new thing called the Internet, identity theft has become a worldwide
problem. For this reason, it is paramount to keep important information such as addresses and telephone numbers
as private as possible when online. If not, you might have thieves coming to your home and stealing your polynomial
identities.
Sure, your identity should be kept safe too, but we were talking more about polynomial identities. A polynomial
identity is just a true equation, often generalized so that it can apply to more than one situation. For instance, (x2 +
y2)2 = (x2 – y2)2 + (2xy)2 is true for any x and y. But how can we be sure that’s true unless we prove it?
Students should be able to prove identities by showing that one side of an equation is equal to the other. That just
takes the same skills they use to organize equations and expressions. They can leave their room-cleaning skills at
home, since we all know how organized those are.
To prove polynomial identities, students can either work from one side of the equation to try to derive the other side,
or can work from both sides to get the same thing. Both are just fine as long as both sides end up being exactly the
same. The goal is to make both sides identical.
Students should also understand the use of certain identities. Sometimes, memorizing an identity is quicker than
working out the algebra longhand, and other times the identity applies to a particular context.
We can prove the identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 by manipulating the equation step by step, showing our work
as we go. We’ll change only one side of the equation and see if we can change it into the other side.
(x2 + y2)2
= (x2 + y2)(x2 + y2)
= x4 + x2y2 + y2x2 + y4
= x4 + 2x2y2 + y4
= x4 – 2x2y2 + 4x2y2 + y4
= (x4 – 2x2y2 + y4) + 4x2y2
= (x2 – y2)2 + (2xy)2
There. Since both sides are equal, we can call this equation an identity. The identity is true for all values of x and y.
For x = 4 and y = 2? Yep. How about for x = -1 and y = 398,128? You betcha.
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.APR.4
Prove polynomial identities and use them to describe numerical
relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 +
(2xy)2 can be used to generate Pythagorean triples.
DESCRIPTION
A.APR.4 Understand that polynomial identities include but are not limited to the product of the sum and difference
of two terms, the difference of two squares, the sum and difference of two cubes, the square of a binomial, etc .
A.APR.4 Prove polynomial identities by showing steps and providing reasons.
A.APR.4 Illustrate how polynomial identities are used to determine numerical relationships such as 25²=(20 + 5)² =
20² + 2 · 5 + 5²
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 41
MATHEMATICS
ESSENTIAL
QUESTION(S)
• How can proving polynomial identities help solve problems efficiently?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.3. Construct viable arguments and critique the reasoning of others.
HS.MP.6. Attend to precision.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Define the Binomial Theorem and compute
combinations.
Apply the Binomial theorem to expand
(x+y)n, when n is a positive integer and
x and y are any number, rather than
expanding by multiplying.
Explain the connection between
Pascal’s Triangle and the
determination of the coefficients
in the expansion of (x+y)n, when
n is a positive integer and x and y
are any number.
EXPLANATIONS
AND EXAMPLES
The Binomial Theorem and Pascal’s Triangle are very important concepts for two reasons. First, they’re capitalized,
and when words are capitalized, you know they’re a big deal. Second, both the Binomial Theorem and Pascal’s
Triangle make expanding polynomials way easier.
Students should be able to expand (x + y)n using Pascal’s Triangle, where x and y can be anything. Pascal’s Triangle
provides the coefficients for the expansion, and the Binomial Theorem explains Pascal’s Triangle. They go together
like ramma lamma lamma ke ding a de dinga dong…whatever that means.
The Binomial Theorem can be explained using combinations, usually because they’re the easiest to visualize and
explain. For example, flipping a coin is a classic example since there are two sides.
How many combinations are possible with 1 heads and 2 tails? We could have HTT, THT, and TTH. There are three
possible combinations. We can assign values to our total number of trials (n = 3) and our total number of desired
outcome (heads, for example, would be k = 1).
The notation nCk is a way of saying, “The number of different combinations we can have if, out of n trials, our
desired outcome occurs k times.” For example, we can use nCk if we flip a coin n times and we want to know how
many different ways we can have k heads (or k tails, whichever we want). We can calculate nCk using the formula
Those exclamation points aren’t just because we’re excited about n and k. They’re mathspeak for “factorial,” which
translates to n × (n – 1) × (n – 2) × … × 2 × 1. But you can be excited about n and k too.
In our example of three coin flips (n = 3) and 1 heads (k = 1), this equals
There are 3 possible ways to get 1 heads out of 3 coin flips.
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.APR.5
(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n, where x and y are any numbers,
with coefficients determined for example by Pascal’s Triangle.
DESCRIPTION
A.APR.5 For small values of n, use Pascal’s Triangle to determine the coefficients of the binomial expansion.
A.APR.5 Use the Binomial Theorem to find the nth term in the expansion of a binomial to a positive power.
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42 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
We can use this same logic when expanding (x + y)n. How many ways can we get 3 x’s and 2 y’s, for example? (Note
that there are the same number of ways to get 2 x’s and 3 y’s.) This can be written as 5C3, which equals
Let’s now think about all ways of getting k x’s and (n – k) y’s out of n outcomes.
When n = 0, we can have 0C0, which equals 1. Our values for n = 2 are 1C0 and 1C1, both of which equal 1 as well. For
n = 2, 2C0 and 2C2 equal 1, but 2C1 equals 2. Students should know that continuing with these calculations would
get us Pascal’s Triangle, in which each number is the sum of the two numbers above it.
Let’s use Pascal’s Triangle to expand (x + y)5.
(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5
The coefficients of each term correspond to the row of Pascal’s Triangle in which n = 5. The exponents of the x’s
start from n and decrease to 0, while the orders of the y’s start at 0 and increase until we get to n. Pretty awesome,
right?
(Source: www.shmoop.com)
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 43
MATHEMATICS
CLUSTER 4. Rewrite rational expressions.
BIG IDEA
• The operations of addition, subtraction and multiplication can be applied to polynomials.
• Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
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44 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• What strategy can be used to rewrite rational expressions in different forms?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Use inspection to rewrite simple rational
expressions in different forms; write a(x)/
b(x) in the form q(x) + r(x)/b(x), where a(x),
b(x), q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of b(x).
Use long division to rewrite simple rational
expressions in different forms; write a(x)/
b(x) in the form q(x) + r(x)/b(x), where a(x),
b(x), q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of b(x).
Use a computer algebra system to rewrite
complicated rational expressions in
different forms; write a(x)/b(x) in the form
q(x) + r(x)/b(x), where a(x), b(x), q(x), and
r(x) are polynomials with the degree of r(x)
less than the degree of b(x).
EXPLANATIONS
AND EXAMPLES
The polynomial q(x) is called the quotient and the polynomial r(x) is called the remainder. Expressing a rational
expression in this form allows one to see different properties of the graph, such as horizontal asymptotes.
This standard is like a Rube Goldberg machine. It looks far more complicated than it really is.
It all boils down to dividing polynomials and expressing the answer properly. Students should have already touched
division and fractions at least a little, having been through the third grade and all. In a way, it’s just a continuation of
the Remainder Theorem, so we recommend covering that first.
Students should already know how to divide polynomials by factoring or long division. As with many divisions, they
won’t all be perfect and a remainder will be left over. Instead of just writing what the remainder is, we now expect
students to actually do something with it.
Let’s say we’re dividing a(x) by b(x), and our answer is q(x) with remainder r(x). Just like the Remainder Theorem, if r(x)
= 0, then b(x) is a factor of a(x). We know that already.
But what if b(x) doesn’t divide a(x) with remainder 0? Well, just like simplifying 13⁄4 to 3 with a remainder of 1, or 3¼,
we can write a(x)⁄b(x) as q(x) with remainder r(x), or r(x)⁄b(x). Just like a remainder of 1 divided by 4 means ¼, a remainder of
r(x) divided by b(x) will give us r(x)⁄b(x).
STANDARD AND DECONSTRUCTION
A.APR.6
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the
form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of b(x), using inspection, long division,
or, for the more complicated examples, a computer algebra system.
DESCRIPTION AA.APR.6 Rewrite rational expressions in the form by using factoring, long division, or
synthetic division. Use a computer algebra system for complicated examples to assist with building a broader
conceptual understanding.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 45
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
All the talk about the degree of r(x) being less than the degree of b(x) just means that r(x) should be “smaller”
than b(x). It wouldn’t make sense to split 13⁄4 into 2 5⁄4 because we can still divide 5 by 4. It’s the same idea, only
polynomial-style.
We suggest relating these polynomial quotients to fractions of integers so that students don’t feel overwhelmed.
It’s understandable for them to be confused when we throw seven different functions at them, but they’ll be a lot
more receptive when they’re working with concepts they already know.
Students should also not be afraid of the big bad long division monster. Often, factoring is near impossible to
figure out when remainders are involved and there are times when synthetic division just won’t cut it. Students
should give in and embrace long division and their lives will be better for it.
(Source: www.shmoop.com)
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46 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How is adding, subtracting, multiplying and dividing of rational expressions similar to that of rational numbers?
MATHEMATICAL
PRACTICE(S)
HS.MP.7. Look for and make use of structure.
HS.MP.8. Look for and express regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Add, subtract, multiply, and divide
rational expressions.
Informally verify that rational
expressions form a system
analogous to the rational
numbers, closed under addition,
subtraction, multiplication, and
division by a nonzero rational
expression.
EXPLANATIONS
AND EXAMPLES
A rational expression is any polynomial divided by any polynomial except for zero. If your students don’t already
know, dividing by zero is catastrophic. In fact, many esteemed mathematicians believe the apocalypse will result
from dividing by zero.
Students should understand that rational expressions are closed under addition, subtraction, multiplication, and
division, meaning that:
• A rational expression plus a rational expression is a rational expression.
• A rational expression minus a rational expression is a rational expression.
• A rational expression times a rational expression is a rational expression.
• A rational expression divided by a rational expression is a rational expression.
It’s only rational.
Students should also know how to add, subtract, multiply, and divide rational expressions. They’re really not that
different from adding, subtracting, and multiplying polynomials. We’re just throwing division in there, too.
Let’s denote a rational expression as
We could worry about remainders here, but keeping rational expressions, as fractions will help us out in the long
run. In fact, rational expressions and fractions are practically identical when it comes to adding, subtracting,
multiplying, and dividing.
If we want to add or subtract 1⁄4 and 2⁄5, we first have to find a common denominator. Only after they both have a
denominator of 5 × 4 = 20 can we add and subtract them. We don’t need common denominators to multiply 1⁄4
and 2⁄5, though. Just multiply the numerators together and multiply the denominators together. When division is
involved, it’s easier to just flip the second fraction and then multiply, so 1⁄4 ÷ 2⁄5 becomes 1⁄4 × 5⁄2.
STANDARD AND DECONSTRUCTION
A.APR.7
(+) Understand that rational expressions form a system analogous to the
rational numbers, closed under addition, subtraction, multiplication, and
division by a nonzero rational expression; add, subtract, multiply, and
divide rational expressions.
DESCRIPTION
A.APR.7 Simplify rational expressions by adding, subtracting, multiplying, or dividing.
A.APR.7 Understand that rational expressions are closed under addition, subtraction, multiplication, and division
(by a nonzero expression).
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 47
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
\Well guess what. Rational expressions are exactly the same, only the integers are replaced with polynomials.
Basically, the rules simplify to these:
In each case, we end up with a polynomial divided by a polynomial, which is a rational expression. Since
polynomials can be defined anywhere, we can’t have any x under the radical. As long as students know how to
multiply and factor polynomials, they should be fine with rational expressions.
(Source: www.shmoop.com)
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MATHEMATICS
CREATING
EQUATIONS
(A-CED)
DOMAIN:
HIGH SCHOOL
ALGEBRA
50 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
DOMAIN Domain: Creating Equations (A-CED) («)
CLUSTERS 1. Create equations that describe numbers or relationships.
ACADEMIC
VOCABULARY
Algebra I
absolute value, complement of an event, compound, conjunction, direct and inverse variation, disjunction,
domain & range, exponential growth (and decay), interest (simple and compound), irrational numbers, joint and
conditional probability, law of large numbers, mathematical model, measure of spread (range, interquartile range),
midpoint formula, outlier, parent function, Pascal’s triangle, polynomial (binomial, trinomial), quadratic formula
(including discriminant), quantitative and qualitative data, radicand, rational expression, real number properties,
real roots (zeros, solutions, x-intercepts), relative frequency, sequences (arithmetic, geometric, Fibonacci),
simulations, subsets of real numbers
Algebra II
amplitude, asymptote, binomial theorem, combination, common ratio (geometric sequence), complete the
square, complex conjugate, complex number, composition (of functions), conic sections (circles, parabola, ellipse,
hyperbola), empirical rule, factorial, focus (pl. foci), independent and dependent events, inverse of a relation,
logarithm, normal distribution, period, permutation, piece-wise function, radian measure, rational function,
regression equation, series (arithmetic, geometric, finite, infinite, etc.), sigma, standard deviation, step function,
synthetic division, transcendental function, trigonometric function, trigonometric identity, unit circle, variance
CLUSTER 1. Create equations that describe numbers or relationships.
BIG IDEA
• Real-world problems are described by creating and solving equations and inequalities.
• Equations describe numbers and relationships.
• Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence
ESSENTIAL
QUESTION(S)
• How are equations used to describe numbers or relationships and solve problems?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Solve linear and exponential equations in
one variable.
Solve inequalities in one variable.
Solve all available types of equations and
inequalities, including root equations and
inequalities, in one variable.
Describe the relationships between the
quantities in the problem (for example,
how the quantities are changing or
growing with respect to each other);
express these relationships using
mathematical operations to create an
appropriate equation or inequality to solve.
Use all available types of
functions to create such
equations, including root
functions, but constrain to simple
cases.
Create equations and inequalities
in one variable to model real-
world situations.
Create equations (linear and
exponential) and inequalities in
one variable and use them to
solve problems.
EXPLANATIONS
AND EXAMPLES
Equations can represent real-world and mathematical problems. Include equations and inequalities that arise
when comparing the values of two different functions, such as one describing linear growth and one describing
exponential growth.
Students should be able to interpret word problems and form equations and inequalities in order to solve the
problem. That means translating a word problem to an algebraic equation.
Let’s be real, here. Math is another language, just like Spanish, Japanese, or Icelandic. When you
start learning a language, you don’t start by translating words like “absquatulate” or “loquacious” or
“pneumonoultramicroscopicsilicovolcanoconiosis” (and yes, that is a real word).
It’s better to start easier, with words like “cat” and “girl” and slowly work your way up. Just the same, if you use simple
linear equations that are familiar to students, they can focus on the translation process and it’ll all go a lot smoother.
Translation is a useful analogy in and of itself because it emphasizes that the algebraic equation is the same as the
word problem, just presented in a different way. In addition to helping students to understand the process, the
translation analogy can also help reassure struggling learners and encourage practice.
After they’ve gotten a hang of the basics, students can start learning quadratic, rational, and exponential functions
to address all aspects of this standard. Once students are familiar with these operations individually, they should be
asked to distinguish them from each another.
As students gain experience, there are additional strategies that should be introduced. One experienced problem
solver strategy is to read the question twice before beginning. It’s a useful piece of advice in general, actually.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 51
MATHEMATICS
ESSENTIAL
QUESTION(S)
• How are equations used to describe numbers or relationships and solve problems?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Solve linear and exponential equations in
one variable.
Solve inequalities in one variable.
Solve all available types of equations and
inequalities, including root equations and
inequalities, in one variable.
Describe the relationships between the
quantities in the problem (for example,
how the quantities are changing or
growing with respect to each other);
express these relationships using
mathematical operations to create an
appropriate equation or inequality to solve.
Use all available types of
functions to create such
equations, including root
functions, but constrain to simple
cases.
Create equations and inequalities
in one variable to model real-
world situations.
Create equations (linear and
exponential) and inequalities in
one variable and use them to
solve problems.
EXPLANATIONS
AND EXAMPLES
Equations can represent real-world and mathematical problems. Include equations and inequalities that arise
when comparing the values of two different functions, such as one describing linear growth and one describing
exponential growth.
Students should be able to interpret word problems and form equations and inequalities in order to solve the
problem. That means translating a word problem to an algebraic equation.
Let’s be real, here. Math is another language, just like Spanish, Japanese, or Icelandic. When you
start learning a language, you don’t start by translating words like “absquatulate” or “loquacious” or
“pneumonoultramicroscopicsilicovolcanoconiosis” (and yes, that is a real word).
It’s better to start easier, with words like “cat” and “girl” and slowly work your way up. Just the same, if you use simple
linear equations that are familiar to students, they can focus on the translation process and it’ll all go a lot smoother.
Translation is a useful analogy in and of itself because it emphasizes that the algebraic equation is the same as the
word problem, just presented in a different way. In addition to helping students to understand the process, the
translation analogy can also help reassure struggling learners and encourage practice.
After they’ve gotten a hang of the basics, students can start learning quadratic, rational, and exponential functions
to address all aspects of this standard. Once students are familiar with these operations individually, they should be
asked to distinguish them from each another.
As students gain experience, there are additional strategies that should be introduced. One experienced problem
solver strategy is to read the question twice before beginning. It’s a useful piece of advice in general, actually.
STANDARD AND DECONSTRUCTION
A.CED.1
Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
DESCRIPTION A.CED.1 Create linear, quadratic, rational and exponential equations and inequalities in one variable and use them
in a contextual situation to solve problems.
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52 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
Writing a list of what is known and a list of what needs to be calculated is also an excellent strategy. Such lists
are especially useful when sorting out unnecessary information, identifying an appropriate formula to utilize, or
constructing a proof. These strategies should be suggested and shown to students after they are proficient with
the basic translation process.
To start off, the chart below may be presented as a dictionary to support word to symbol translation. Students can
also add to the chart as they find other key words or phrases.
(Source: www.shmoop.com)
Algebra Symbol Key Words
= equals
1. all
2. equals
3. gives
4. is, are, was, were, will be
5. results
6. same
7. yields
< is less than 1. below
2. less than
≤ is less than or equal to 1. maximum of
2. not more than
> is greater than
1. greater than
2. more than
3. over
≥ is greater than or equal to
1. at least
2. minimum of
3. not less than
+ addition
1. add
2. and
3. combine
4. increase
5. more
6. plus
7. raise
8. sum
9. together
10. total
– subtraction
1. decrease
2. difference
3. fewer
4. less
5. lose
6. minus
7. reduce
× multiplication
1. directly proportional
2. double(× 2), triple(× 3), etc.
3. group of
4. linear
5. multiplied
6. product
7. times
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 53
MATHEMATICS
Algebra Symbol Key Words
/ division
1. average
2. cut
3. divided by/into
4. each
5. inversely proportional
6. out of
7. per
8. pieces
9. quotient
10. ratio
12. share
13. split
xn power 1. power
2. square (n = 2), cube (n = 3), etc.
nx exponential
1. decays
2. doubles (n = 2), triples (n = 3), quadruples (n = 4), etc.
3. grows
4. rate of n per x
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(Source: www.shmoop.com)
http://www.shmoop.com
54 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How are equations used to describe numbers or relationships and solve problems?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Identify the quantities in a
mathematical problem or real-
world situation that should be
represented by distinct variables
and describe what quantities the
variables represent.
Graph one or more created
equations on coordinate axes with
appropriate labels and scales.
Create at least two equations in two or
more variables to represent relationships
between quantities.
Justify which quantities in a
mathematical problem or real-
world situation are dependent and
independent of one another and which
operations represent those relationships.
Determine appropriate units for the
labels and scale of a graph depicting the
relationship between equations created
in two or more variables.
EXPLANATIONS
AND EXAMPLES
This standard has two significant components. The first is translating word problems into equations with two or
more variables. The more the merrier. Well, maybe not in this case. Translating word problems to create simple
equations with two or more variables is not that different conceptually from creating equations with one variable.
The main difference is that more complicated mathematical relationships such as systems of equations, functions,
and proportions may develop (along with nausea, headaches, and spontaneous yodeling). In any case, this aspect of
this standard should be taught with the previous one.
The second component is creating graphs of equations on coordinate axes, which incorporates multiple skills such
as visual perception, interpreting data, and synthesizing information. Such graphs relate to equations with multiple
equations by relating one variable to another.
Take lines, for example. In the form y = mx + b, we can look at either x or y and any defined value for x will give us a
defined value for y, and vice versa. Graphs can help visualize these relationships between variables and facilitate the
connection of equations to the graphs that represent them. Yearnin’ for more graphin’? Don’t worry. There’ll be more
down the line.
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.CED.2
Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels and
scales.
DESCRIPTION
A.CED.2 Create equations in two or more variables to represent relationships between quantities.
A.CED.2 Graph equations in two variables on a coordinate plane and label the axes and scales.
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 55
MATHEMATICS
ESSENTIAL
QUESTION(S)
• How are equations used to describe numbers or relationships and solve problems?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Recognize when a modeling
context involves constraints.
Interpret solutions as viable or nonviable
options in a modeling context.
Determine when a problem should be
represented by equations, inequalities,
systems of equations and/ or inequalities.
Represent constraints by equations or
inequalities, and by systems of equations
and/or inequalities.
EXPLANATIONS
AND EXAMPLES
Students have already translated words into algebraic equations (sometimes with more than one variable!) and have
actually taken the time to solve the problem. We hate to make you the villain here, but you have to tell them their
work isn’t over. Students now have to interpret the results. This standard is about one thing: analysis. Well… actually
it’s about three things:
1. Creating equations/inequalities or systems of equations/inequalities.
2. Solving these equations/inequalities or system of equations.
3. Interpreting the answer properly.
To analyze problems in which multiple relationships affect multiple variables, students must be able to create
systems of equations, solve them, and interpret the results appropriately.
Creating Systems of Equations
In order to create systems of equations from word problems or other contexts, students need to be able to
differentiate the relations and create equations for each. To support this, should already be able to create equations
from word problems.
Creating equations from a word problem or similar context is a three-step translation process:
1. Translate the equality or inequality. (=, <, >, ≤, or ≥)
2. Translate the operations. (+, –, ×, ÷, xn, nx)
3. Translate the numbers and variables.
Systems of equations are identified during step one of this process. Students need to be able to read a problem and
identify how many equality and inequality relations are described. Then, they should write each down separately.
STANDARD AND DECONSTRUCTION
A.CED.3
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 context. For example, represent
inequalities describing nutritional and cost constraints on combinations of
different foods.
DESCRIPTION
A.CED.3 Write and use a system of equations and/or inequalities to solve a real world problem. Recognize that the
equations and inequalities represent the constraints of the problem. Use the Objective Equation and the Corner
Principle to determine the solution to the problem. (Linear Programming)
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56 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
Once these are written down, students perform steps two and three (translating the operations, numbers, and
variables) independently for each equation. They should also simplify each equation individually before working
to solve the system of equations.
Solving Systems of Equations
Solving systems of equations can be done through substitution or adding the two equations together to cancel
out one of the variables. The goal is to eliminate one variable so that we can find the solution for the other and
then substitute that answer back in to find the value of the second variable.
Hopefully, students already know how to solve systems of equations. After all, it’s necessary in order to interpret
results from a set of equations.
Interpreting Results from Systems of Equations
In many situations, students struggle to understand what an algebraic result means in the context of a word
problem. This is especially true when systems of equations are involved and when they arrive at solutions that are
correct algebraically, but incorrect in context.
For instance, a question about how many tops hats a giraffe can wear might produce the number 6.25 as the
answer. This might make sense algebraically, but in the context of giraffes wearing top hats—how can a giraffe
wear one fourth of a hat? The logical answer would then have to be 6 (although logic might not be our biggest
concern if we’re talking about giraffes in top hats).
Such algebraic solutions also present a challenge when multiple roots are encountered or when cancelling
rational expressions. Describing what values of a variable are allowed when recording the variable information is
one strategy for dealing with this problem. (For instance, we could write that giraffes only wear top hats in whole
numbers.)
A common error is to report an answer based on a different variable in the problem. Recording the variable
information helps to prevent such errors. If this is a common issue, students should try highlighting the quantity of
interest in the problem, the matching variable name, and the eventual result. They can then check all highlighted
items to ensure that they match.
Multiple variables present even more of a challenge, as students often need to find information for more than
one variable in a problem. Highlighting the different information requested in different colors is one strategy.
Another possibility is to have students solve for all variables before interpreting a solution; however, this becomes
troublesome in more complex problems.
Interpreting results should be performed throughout the algebra curriculum. With enough practice, students will
perform whatever strategies work best for them naturally.
(Source: www.shmoop.com)
ESSENTIAL
QUESTION(S)
• How are equations used to describe numbers or relationships and solve problems?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Define a quantity of interest to
mean any numerical or algebraic
quantity (e.g. , 2(a/b)=d in which 2
is the quantity of interest showing
that d must be even; (πr²h/3)=V
and πr2h=V showing that V =3* V )
Rearrange formulas to highlight a
quantity of interest, using the same
reasoning as in solving equations. (e.g. π
r² can be re-written as (π r)r which makes
the form of this expression resemble Bh.
The quantity of interest could also be =
+ +…+
EXPLANATIONS
AND EXAMPLES
Students can rearrange formulas until their pencils whittle down to toothpicks, but it won’t help them one bit if
the formulas they use won’t enable them actually solve the problem. That means students should be able to match
commonly encountered formulas to context in word problems as well as rearranging them to solve for whatever
value they want.
Matching Formulas to Create Equations
In addition to being able to translate word problems into equations, students also need to be able to identify when
a common formula is needed for the given context. These are most commonly geometric formulas (like perimeter,
area, or volume of various shapes) or physical formulas (such as F = ma, p = mv, V = IR, v = d⁄t, KE = ½mv2, or GPE =
mgh).
It is important to note that this assumes that students are already familiar with the relevant formulas from previous
learning. Students who are not already familiar with the formulas need to be supported in understanding them
before they will be able to match them to contexts. Matching formulas may be treated as a variation from the
creating equations process. When attempting to write down the equality or inequality, students will notice that there
isn’t enough information. A general relationship might be implied, but no specific equality or inequality is described.
What ever will they do?
Well, students need to identify the formula that describes that relationship. They can look for clues such as
appropriate units (volume, for example, is always in cubic units). Once the correct formula is matched to the
contextual relationship, the students can continue solving the problem as usual.
Rearranging Formulas
Once a formula is written algebraically, students can manipulate it however they want (within mathematical reason).
The process for rearranging it is identical to the process of rearranging any equation or system of equations. It
involves simplifying expressions and solving equations, which students should already know how to do.
The best way to rearrange questions when looking for a particular value is to isolate that value. Students should
rearrange the equations so that the desired value is on one side of the equal sign, and there’s a whole big mess of
stuff on the other. That way, they’ll be able to plug in the values they know and end up with whatever they want
equals some number.
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.CED.4
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.
DESCRIPTION A.CED.4 Solve multi-variable formulas or literal equations, for a specific variable.
http://www.shmoop.com
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 57
MATHEMATICS
ESSENTIAL
QUESTION(S)
• How are equations used to describe numbers or relationships and solve problems?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Define a quantity of interest to
mean any numerical or algebraic
quantity (e.g. , 2(a/b)=d in which 2
is the quantity of interest showing
that d must be even; (πr²h/3)=V
and πr2h=V showing that V =3* V )
Rearrange formulas to highlight a
quantity of interest, using the same
reasoning as in solving equations. (e.g. π
r² can be re-written as (π r)r which makes
the form of this expression resemble Bh.
The quantity of interest could also be =
+ +…+
EXPLANATIONS
AND EXAMPLES
Students can rearrange formulas until their pencils whittle down to toothpicks, but it won’t help them one bit if
the formulas they use won’t enable them actually solve the problem. That means students should be able to match
commonly encountered formulas to context in word problems as well as rearranging them to solve for whatever
value they want.
Matching Formulas to Create Equations
In addition to being able to translate word problems into equations, students also need to be able to identify when
a common formula is needed for the given context. These are most commonly geometric formulas (like perimeter,
area, or volume of various shapes) or physical formulas (such as F = ma, p = mv, V = IR, v = d⁄t, KE = ½mv2, or GPE =
mgh).
It is important to note that this assumes that students are already familiar with the relevant formulas from previous
learning. Students who are not already familiar with the formulas need to be supported in understanding them
before they will be able to match them to contexts. Matching formulas may be treated as a variation from the
creating equations process. When attempting to write down the equality or inequality, students will notice that there
isn’t enough information. A general relationship might be implied, but no specific equality or inequality is described.
What ever will they do?
Well, students need to identify the formula that describes that relationship. They can look for clues such as
appropriate units (volume, for example, is always in cubic units). Once the correct formula is matched to the
contextual relationship, the students can continue solving the problem as usual.
Rearranging Formulas
Once a formula is written algebraically, students can manipulate it however they want (within mathematical reason).
The process for rearranging it is identical to the process of rearranging any equation or system of equations. It
involves simplifying expressions and solving equations, which students should already know how to do.
The best way to rearrange questions when looking for a particular value is to isolate that value. Students should
rearrange the equations so that the desired value is on one side of the equal sign, and there’s a whole big mess of
stuff on the other. That way, they’ll be able to plug in the values they know and end up with whatever they want
equals some number.
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.CED.4
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.
DESCRIPTION A.CED.4 Solve multi-variable formulas or literal equations, for a specific variable.
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REASONING
WITH EQUATIONS
AND INEQUALITIES
(A-REI)
DOMAIN:
HIGH SCHOOL
ALGEBRA
MATHEMATICS
60 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
DOMAIN Reasoning with Equations and Inequalities (A-REI)
CLUSTERS
1. Understand solving equations as a process of reasoning and explain the reasoning.
2. Solve equations and inequalities in one variable.
3. Solve systems of equations.
4. Represent and solve equations and inequalities graphically.
ACADEMIC
VOCABULARY
Algebra I
absolute value, complement of an event, compound, conjunction, direct and inverse variation, disjunction,
domain & range, exponential growth (and decay), interest (simple and compound), irrational numbers, joint and
conditional probability, law of large numbers, mathematical model, measure of spread (range, interquartile range),
midpoint formula, outlier, parent function, Pascal’s triangle, polynomial (binomial, trinomial), quadratic formula
(including discriminant), quantitative and qualitative data, radicand, rational expression, real number properties,
real roots (zeros, solutions, x-intercepts), relative frequency, sequences (arithmetic, geometric, Fibonacci),
simulations, subsets of real numbers
Algebra II
amplitude, asymptote, binomial theorem, combination, common ratio (geometric sequence), complete the
square, complex conjugate, complex number, composition (of functions), conic sections (circles, parabola, ellipse,
hyperbola), empirical rule, factorial, focus (pl. foci), independent and dependent events, inverse of a relation,
logarithm, normal distribution, period, permutation, piece-wise function, radian measure, rational function,
regression equation, series (arithmetic, geometric, finite, infinite, etc.), sigma, standard deviation, step function,
synthetic division, transcendental function, trigonometric function, trigonometric identity, unit circle, variance
CLUSTER 1. Understand solving equations as a process of reasoning and explain the reasoning.
BIG IDEA
• Reasoned explanations are required to solve equations and inequalities.
• Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
• Relationships between two sets of numbers can be described by mathematical rules, where a
function is a unique rule that has a one-to-one correspondence.
ESSENTIAL
QUESTION(S)
• What are the steps and strategies to justify a solution to a problem?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Know that solving an equation means
that the equation remains balanced
during each step.
Recall the properties of equality.
Explain why, when solving equations,
it is assumed that the original equation
is equal.
Determine if an equation has a
solution.
Choose an appropriate method for
solving the equation.
Justify solution(s) to equations by
explaining each step in solving a
simple equation using the properties
of equality, beginning with the
assumption that the original equation
is equal.
Construct a mathematically viable
argument justifying a given, or self-
generated, solution method.
EXPLANATIONS
AND EXAMPLES
Properties of operations can be used to change expressions on either side of the equation to equivalent expressions.
In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant
produces an equation with the same solutions. Other operations, such as squaring both sides, may produce
equations that have extraneous solutions.
Who was it that said, “A journey of a thousand miles begins with a single step”? Was it Socrates? Confucius? Bugs
Bunny?
Whoever it was, they probably weren’t thinking about algebra when they coined that gem. To many students,
solving an algebraic equation may feel like a journey of a thousand miles. Before students begin with their single
step, they should probably get a compass or something.
Students should be able to figure out the logical next step in solving an equation using the previous step. Sounds
simple, but it takes more than just putting one foot in front of the other. Knowing that whatever is done to one side
of the equation must be done to the other is a good start, but that doesn’t tell them what to do.
Ideally, students should not be memorizing a set of rules and procedures and using them to solve equations. They
should understand how the next step in solving an equation can be logically derived from the previous step, but
there is a general format for how to get started on their algebraic journey.
When we solve an equation, it’s usually a good idea to get the variable we want on one side of the equal sign. Then,
we do what we can to simplify it as much as we can.
STANDARD AND DECONSTRUCTION
A.REI.1
Explain each step in solving a simple equation as following from the
equality of numbers asserted at the previous step, starting from the
assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
DESCRIPTION
A.REI.1 Assuming an equation has a solution, construct a convincing argument that justifies each step in the
solution process. Justifications may include the associative, commutative, and division properties, combining like
terms, multiplication by 1, etc.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 61
MATHEMATICS
ESSENTIAL
QUESTION(S)
• What are the steps and strategies to justify a solution to a problem?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Know that solving an equation means
that the equation remains balanced
during each step.
Recall the properties of equality.
Explain why, when solving equations,
it is assumed that the original equation
is equal.
Determine if an equation has a
solution.
Choose an appropriate method for
solving the equation.
Justify solution(s) to equations by
explaining each step in solving a
simple equation using the properties
of equality, beginning with the
assumption that the original equation
is equal.
Construct a mathematically viable
argument justifying a given, or self-
generated, solution method.
EXPLANATIONS
AND EXAMPLES
Properties of operations can be used to change expressions on either side of the equation to equivalent expressions.
In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant
produces an equation with the same solutions. Other operations, such as squaring both sides, may produce
equations that have extraneous solutions.
Who was it that said, “A journey of a thousand miles begins with a single step”? Was it Socrates? Confucius? Bugs
Bunny?
Whoever it was, they probably weren’t thinking about algebra when they coined that gem. To many students,
solving an algebraic equation may feel like a journey of a thousand miles. Before students begin with their single
step, they should probably get a compass or something.
Students should be able to figure out the logical next step in solving an equation using the previous step. Sounds
simple, but it takes more than just putting one foot in front of the other. Knowing that whatever is done to one side
of the equation must be done to the other is a good start, but that doesn’t tell them what to do.
Ideally, students should not be memorizing a set of rules and procedures and using them to solve equations. They
should understand how the next step in solving an equation can be logically derived from the previous step, but
there is a general format for how to get started on their algebraic journey.
When we solve an equation, it’s usually a good idea to get the variable we want on one side of the equal sign. Then,
we do what we can to simplify it as much as we can.
STANDARD AND DECONSTRUCTION
A.REI.1
Explain each step in solving a simple equation as following from the
equality of numbers asserted at the previous step, starting from the
assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
DESCRIPTION
A.REI.1 Assuming an equation has a solution, construct a convincing argument that justifies each step in the
solution process. Justifications may include the associative, commutative, and division properties, combining like
terms, multiplication by 1, etc.
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62 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
For example, let’s solve the equation x2 – 3x – 7 = 2x + 17. Subtracting 2x + 17 from both sides gets all our x’s on
one side of the equation. That’s a good place to start.
x2 – 3x – 7 – (2x + 17) = 2x + 17 – (2x + 17)
x2 – 5x – 24 = 0
We also took the opportunity to simplify by combining like terms. Since the equation we have is a quadratic
equation, we can factor it into the product of two linear terms.
(x – 8)(x + 3) = 0
This product will equal 0 when either x – 8 or x + 3 is equal to 0, so we can solve it by setting each of them to 0. As
a result, x = 8 and -3.
If students are really struggling, we suggest starting simple. Give them easier linear equations and slowly work
your way to more complex ones. Point out patterns in equations, make sure they know the quadratic formula, and
remind them of helpful factoring tricks.
Most of all, get them to practice. It’s difficult to take even the first step of a journey if you can’t walk. After enough
exercises, your students should be able to tackle any thousand-mile journey faster than the Road Runner.
(Source: www.shmoop.com)
ESSENTIAL
QUESTION(S)
• What are the steps and strategies to justify a solution to a problem?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.3. Construct viable arguments and critique the reasoning of others.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Determine the domain of a rational
function.
Determine the domain of a radical
function.
Solve radical equations in one variable.
Solve rational equations in one
variable.
Give examples showing how
extraneous solutions may arise when
solving rational and radical equations.
EXPLANATIONS
AND EXAMPLES
Rational equations might sometimes be a bit more radical than students would like. Those square root signs can instill
terror in the heart of any student unprepared for them. If students overcame their fear of monsters under the bed (and
hopefully they did), they’ll get over their fear of radicals too.
Rational equations mean that fractions are involved. Radical equations mean that square roots are involved. Students
should know how to deal with both separately and together.
A radical equation is one in which the variable is under the radical sign. When solving radical equations, it’s usually
best to leave the radicals for last unless there’s a quick and easy way to get rid of all of them. If only there were a quick
and easy way to get rid of monsters.
For example, is a radical equation. To solve, students should add 4 to both sides and get
, or . All that’s left is squaring both sides to get x = 256. Not too scary, right?
Of course, solving radical equations means students have to understand how to combine radicals. For example,
since both terms have the same thing under the radical. We cannot combine and this
same way. We could, however, rewrite as in which case . Not as pretty,
but nonetheless doable.
Students should know how to combine, manipulate, and rewrite radical expressions. This usually takes practice and
repetition. When all else fails, tell students to treat radicals like they’d treat variables. This is the one time the Golden
Rule doesn’t apply.
Students should already know how to solve rational equations. They should be able to find x faster than a pirate with
a treasure map. And if he has an eye patch then it shouldn’t even be a contest.
A rational equation might look like
In this case, let’s say it does. First, cross-multiply to get 3(x – 4) = (x – 4)(x + 2). Since we have (x – 4) on both sides,
we can reduce the equation to 3 = x + 2. Our final answer is x = 1.
STANDARD AND DECONSTRUCTION
A.REI.2 Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
DESCRIPTION A.REI.2 Solve simple rational and radical equations in one variable and provide examples of how extraneous
solutions arise.
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COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 63
MATHEMATICS
ESSENTIAL
QUESTION(S)
• What are the steps and strategies to justify a solution to a problem?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.3. Construct viable arguments and critique the reasoning of others.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Determine the domain of a rational
function.
Determine the domain of a radical
function.
Solve radical equations in one variable.
Solve rational equations in one
variable.
Give examples showing how
extraneous solutions may arise when
solving rational and radical equations.
EXPLANATIONS
AND EXAMPLES
Rational equations might sometimes be a bit more radical than students would like. Those square root signs can instill
terror in the heart of any student unprepared for them. If students overcame their fear of monsters under the bed (and
hopefully they did), they’ll get over their fear of radicals too.
Rational equations mean that fractions are involved. Radical equations mean that square roots are involved. Students
should know how to deal with both separately and together.
A radical equation is one in which the variable is under the radical sign. When solving radical equations, it’s usually
best to leave the radicals for last unless there’s a quick and easy way to get rid of all of them. If only there were a quick
and easy way to get rid of monsters.
For example, is a radical equation. To solve, students should add 4 to both sides and get
, or . All that’s left is squaring both sides to get x = 256. Not too scary, right?
Of course, solving radical equations means students have to understand how to combine radicals. For example,
since both terms have the same thing under the radical. We cannot combine and this
same way. We could, however, rewrite as in which case . Not as pretty,
but nonetheless doable.
Students should know how to combine, manipulate, and rewrite radical expressions. This usually takes practice and
repetition. When all else fails, tell students to treat radicals like they’d treat variables. This is the one time the Golden
Rule doesn’t apply.
Students should already know how to solve rational equations. They should be able to find x faster than a pirate with
a treasure map. And if he has an eye patch then it shouldn’t even be a contest.
A rational equation might look like
In this case, let’s say it does. First, cross-multiply to get 3(x – 4) = (x – 4)(x + 2). Since we have (x – 4) on both sides,
we can reduce the equation to 3 = x + 2. Our final answer is x = 1.
STANDARD AND DECONSTRUCTION
A.REI.2 Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
DESCRIPTION A.REI.2 Solve simple rational and radical equations in one variable and provide examples of how extraneous
solutions arise.
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64 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
Students should know that sometimes, algebraic manipulation produces extraneous solutions. For example,
multiplying the fairly simple equation x + 5 = 0 by x will give x2 + 5x = 0. Now, both x = 0 and x = -5 will satisfy that
quadratic. However, looking at x + 5 = 0, we can see that x = 0 won’t work for the original equation (because 0 + 5
≠ 0). That means x = 0 is an extraneous solution.
To check for extraneous solutions, students should plug in their final answers back into the original equation. If the
equation produces an incorrect statement (like 5 = 0), then they’ll know that solution didn’t really exist. Much like
the monsters under the bed…we hope.
(Source: www.shmoop.com)
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COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 65
MATHEMATICS
CLUSTER 2. Solve equations and inequalities in one variable
BIG IDEA • Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
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66 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• What are the steps and strategies to justify a solution to a problem?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.7. Look for and make use of structure.
HS.MP.8. Look for and express regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Recall properties of equality.
Solve multi-step equations in one
variable.
Solve multi-step inequalities in one
variable.
Determine the effect that rational
coefficients have on the inequality
symbol and use this to find the
solution set.
Solve equations and inequalities with
coefficients represented by letters.
EXPLANATIONS
AND EXAMPLES
With linear equations, students should be able to find the solution or solutions that make the equation true. We
usually get one or several specific answers with equations, but inequalities sing a slightly different tune. As different as
Under Pressure and Ice Ice Baby.
Students should first understand the difference between an equation and inequality. An equation uses the = sign
while an equality may use <, >, ≤, or ≥. If we find that x ≤ 2, we know that x can be 2 or anything less than 2. If we
know that x < 2, x cannot be 2, but it can be anything less than 2.
With inequalities, students should find the set of numbers that make the inequality true. Inequalities won’t tell us
exactly which number x will equal. Instead, it’ll give us a range of possible x values, all of which will work for the
inequality.
Students should also know how to work with inequalities. Algebraically, they aren’t that different from an equal sign.
Still, multiplying and dividing by negative numbers switches the direction of the sign (1 > -2 but multiplying both
sides by -1 gives us -1 < 2).
If students are unsure, it might be helpful for them to visualize inequalities on a number line. Sometimes, letters may
represent constants and coefficients in equations. Students should know how to treat these as numbers. For instance,
the answer to the equation x + 4m = 2x + m would be written as x = 3m. It’s okay for our solution to be in terms of m
because m is treated as a constant. Isn’t that nice, nice, baby?
We apologize in advance if that baseline is stuck in your head for the rest of the day.
(Source: www.shmoop.com)
ESSENTIAL
QUESTION(S)
What are the steps and strategies to justify a solution to a problem?
How can the method of completing the square be used to derive the quadratic formula?
How do I determine the most appropriate and efficient strategy to use to solve quadratic equations?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.7. Look for and make use of structure.
HS.MP.8. Look for and express regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
STANDARD AND DECONSTRUCTION
A.REI.3 Solve linear equations and inequalities in one variable, including
equations with coefficients represented by letters.
DESCRIPTION
A.REI.3 Solve linear equations in one variable, including coefficients represented by letters.
A.REI.3 Solve linear inequalities in one variable, including coefficients represented by letters.
SUBSTANDARD
DECONSTRUCTED
A.REI.4a Use the method of completing the square to transform any
quadratic equation in x into an equation of the form (x – p)2 = q that has the
same solutions. Derive the quadratic formula from this form.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Use the method of completing the
square to transform any quadratic
equation in x into an equation of
the form (x-p)2 = q that has the
same solutions.
Solve quadratic equations in one
variable.
Derive the quadratic formula by
completing the square on the
standard form of a quadratic
equation.
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COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 67
MATHEMATICS
ESSENTIAL
QUESTION(S)
What are the steps and strategies to justify a solution to a problem?
How can the method of completing the square be used to derive the quadratic formula?
How do I determine the most appropriate and efficient strategy to use to solve quadratic equations?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.7. Look for and make use of structure.
HS.MP.8. Look for and express regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 T 3 o 4
STANDARD AND DECONSTRUCTION
A.REI.4 Solve quadratic equations in one variable.
DESCRIPTION
A.REI.4a Transform a quadratic equation written in standard form to an equation in vertex form - by completing the
square. (x – p )² = q
A.REI.4a Derive the quadratic formula by completing the square on the standard form of a quadratic equation.
A.REI.4b Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and
completing the square.
A.REI.4b Understand why taking the square root of both sides of an equation yields two solutions.
A.REI.4b Use the quadratic formula to solve any quadratic equation, recognizing the formula produces all complex
solutions. Write the solutions in the form, where a and b are real numbers. a ± bi
A.REI.4b Explain how complex solutions affect the graph of a quadratic equation.
SUBSTANDARD
DECONSTRUCTED
A.REI.4a Use the method of completing the square to transform any
quadratic equation in x into an equation of the form (x – p)2 = q that has the
same solutions. Derive the quadratic formula from this form.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Use the method of completing the
square to transform any quadratic
equation in x into an equation of
the form (x-p)2 = q that has the
same solutions.
Solve quadratic equations in one
variable.
Derive the quadratic formula by
completing the square on the
standard form of a quadratic
equation.
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68 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
SUBSTANDARD
DECONSTRUCTED
A.REI.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking
square roots, completing the square, the quadratic formula and factoring,
as appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Solve quadratic equations by
inspection taking square roots,
completing the square, the
quadratic formula and factoring.
Express complex solutions as a ±
bi for real numbers solutions as a
and b.
Recognize when the quadratic
formula gives complex solutions.
Determine appropriate strategies to
solve problems involving quadratic
equations, as appropriate to the
initial form of the equation.
EXPLANATIONS
AND EXAMPLES
Students should know how to solve equations involving terms with one variable to the second degree. These
equations may be written in any form, the most common being the standard form of a quadratic equation, ax2 +
bx + c = 0. Students should know and apply the three main ways to solve a quadratic equation: stop, drop, and roll.
Oh, wait. Scratch that. We meant factor, complete the square, and the quadratic formula.
Your students should already know what factoring is, and that it’s possible with simple quadratic equations like
x2 + x – 12 = 0. When factoring, students should look for two numbers that add to the coefficient b (in this case, 1)
and multiply to get the constant c (in this case, -12). Easier said than done, unless you’ve got a mouthful of peanut
butter.
For the equation x2 + x – 12, the two numbers that work are -3 and 4. They add to get 1 and multiply to get -12, so
we can factor x2 + x – 12 into (x – 3)(x + 4). Now it’s way easier to solve when we set the equation to equal 0. Since
x – 3 = 0 and x + 4 = 0, our answers are x = 3 and x = -4. Note that this method only works when a is 1.
(a.) Example.
This is a sneaky and fun way to solve unruly quadratic equations. It involves creating and factoring a “perfect
square” (meaning, a quadratic expression whose two linear factors are the same). Let’s solve the same equation as
we did before: x2 + x – 12 = 0.
First, add 12 to both sides: x2 + x = 12. Next, complete the square by making the left side a perfect square. To do
that, we take b, divide by 2, and square the result. In this case, our b = 1, which ends up being ¼. Add this to both
sides of the equation.
The left side is a perfect square because its two linear factors are the same: x + ½. Therefore, we can re-write the
left side of the equation and simplify the right-hand side.
From there, we can square root both sides and subtract both sides by ½.
x = 3, -4
(b) Example.
The quadratic equation is essentially a shortcut to completing the square. It may not seem like a shortcut to your
students, but it beats spending an hour trying to factor an equation that just won’t budge.
Students should know the quadratic formula and how to use it. When given a quadratic equation ax2 + bx + c = 0,
the quadratic formula is this ravishing thing:
EXPLANATIONS
AND EXAMPLES
(continued)
Of course, if all that business under the radical turns out to be negative, we’ve got a real problem. We can’t square
root negative numbers, right? As long as students are familiar with complex numbers and i, this real problem
becomes an imaginary one very quickly. Then we can simplify the rest so that it follows the form a ± bi.
While students may cringe at this monstrous behemoth of a formula, they’ll soon appreciate its sophisticated
and simple beauty. In fact, many of them will memorize it without even realizing it. Armed with those three ways,
students should be able to attack just about any quadratic formula using only a pencil and their bright and shiny
noggins (and maybe a calculator, if you’re feeling generous).
(Source: www.shmoop.com)
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COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 69
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
Of course, if all that business under the radical turns out to be negative, we’ve got a real problem. We can’t square
root negative numbers, right? As long as students are familiar with complex numbers and i, this real problem
becomes an imaginary one very quickly. Then we can simplify the rest so that it follows the form a ± bi.
While students may cringe at this monstrous behemoth of a formula, they’ll soon appreciate its sophisticated
and simple beauty. In fact, many of them will memorize it without even realizing it. Armed with those three ways,
students should be able to attack just about any quadratic formula using only a pencil and their bright and shiny
noggins (and maybe a calculator, if you’re feeling generous).
(Source: www.shmoop.com)
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70 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
CLUSTER 3. Solve systems of equations.
BIG IDEA • Equations and inequalities that can be solved using arithmetic and algebraic rules and equivalence.
ESSENTIAL
QUESTION(S)
• What strategy is used to solve systems of two equations?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.3. Construct viable arguments and critique the reasoning of others.
DOK Range Target
for Instruction &
Assessment
o 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Recognize and use properties of
equality to maintain equivalent
systems of equations.
Justify that replacing one equation in
a two-equation system with the sum
of that equation and a multiple of the
other will yield the same solutions as
the original system.
EXPLANATIONS
AND EXAMPLES
Students should be able to understand how manipulating systems of equations will lead to those equations’ solutions.
Let’s say we have two functions that we’re trying to solve simultaneously. We’ll call our functions f(x, y)and g(x, y),
because they’re funky and groovy. Yeah, we took our functions straight out of the 70s.
For instance, let’s say that f(x, y) is 2x + 4 = 3y and g(x, y) is -x + 8y = 2. We can rewrite these as 2x – 3y + 4 = 0 and x – 8y
+ 2 = 0, in which case f(x, y) = 2x – 3y + 4 and g(x, y) = x – 8y + 2. Now, we have our awesome functions (or…funk-
tions?).
If one solution to both f(x, y) = 0 and g(x, y) = 0 is when x = a and y = b, then f(a, b) = 0 and g(a, b) = 0. That means a and
b must also be the x and y values for the system of equations f(x, y) + g(x, y) = 0 and nf(x, y) = 0, where n is a constant.
After all, f(a, b) + g(a, b) = 0 + 0 = 0, and nf(a, b) = n(0) = 0.
Basically, this means that system of equations f(x, y) = 0 and g(x, y) = 0 will have the same solutions as f(x, y) + g(x, y) = 0
and nf(x, y) = 0.
Students should know that this is useful when solving a system of equations (as in, 2 or more of ‘em) because it means
we can multiply equations by constants and add them together all we want. That’ll help us find the x and y values that
work for all equations in the system.
We strongly urge you to relate this information back to graphing linear and quadratic equations. Explain that the
solution to the system of equations is really the point (or points) where the functions intersect. For example, functions
f(x, y) and g(x, y) intersect at (a, b) since those are the x and y values that make the two equations equal to each other.
It’s all connected, just like we’re all connected. Pretty groovy, right?
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.REI.5
Prove that, given a system of two equations in two variables, replacing one
equation by the sum of that equation and a multiple of the other produces
a system with the same solutions.
DESCRIPTION
A.REI.5 Solve systems of equations using the elimination method (sometimes called linear combinations).
A.REI.5 Solve a system of equations by substitution (solving for one variable in the first equation and substituting it
into the second equation).
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 71
MATHEMATICS
ESSENTIAL
QUESTION(S)
• What strategy is used to solve systems of two equations?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.3. Construct viable arguments and critique the reasoning of others.
DOK Range Target
for Instruction &
Assessment
o 1 T 2 T 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Recognize and use properties of
equality to maintain equivalent
systems of equations.
Justify that replacing one equation in
a two-equation system with the sum
of that equation and a multiple of the
other will yield the same solutions as
the original system.
EXPLANATIONS
AND EXAMPLES
Students should be able to understand how manipulating systems of equations will lead to those equations’ solutions.
Let’s say we have two functions that we’re trying to solve simultaneously. We’ll call our functions f(x, y)and g(x, y),
because they’re funky and groovy. Yeah, we took our functions straight out of the 70s.
For instance, let’s say that f(x, y) is 2x + 4 = 3y and g(x, y) is -x + 8y = 2. We can rewrite these as 2x – 3y + 4 = 0 and x – 8y
+ 2 = 0, in which case f(x, y) = 2x – 3y + 4 and g(x, y) = x – 8y + 2. Now, we have our awesome functions (or…funk-
tions?).
If one solution to both f(x, y) = 0 and g(x, y) = 0 is when x = a and y = b, then f(a, b) = 0 and g(a, b) = 0. That means a and
b must also be the x and y values for the system of equations f(x, y) + g(x, y) = 0 and nf(x, y) = 0, where n is a constant.
After all, f(a, b) + g(a, b) = 0 + 0 = 0, and nf(a, b) = n(0) = 0.
Basically, this means that system of equations f(x, y) = 0 and g(x, y) = 0 will have the same solutions as f(x, y) + g(x, y) = 0
and nf(x, y) = 0.
Students should know that this is useful when solving a system of equations (as in, 2 or more of ‘em) because it means
we can multiply equations by constants and add them together all we want. That’ll help us find the x and y values that
work for all equations in the system.
We strongly urge you to relate this information back to graphing linear and quadratic equations. Explain that the
solution to the system of equations is really the point (or points) where the functions intersect. For example, functions
f(x, y) and g(x, y) intersect at (a, b) since those are the x and y values that make the two equations equal to each other.
It’s all connected, just like we’re all connected. Pretty groovy, right?
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.REI.5
Prove that, given a system of two equations in two variables, replacing one
equation by the sum of that equation and a multiple of the other produces
a system with the same solutions.
DESCRIPTION
A.REI.5 Solve systems of equations using the elimination method (sometimes called linear combinations).
A.REI.5 Solve a system of equations by substitution (solving for one variable in the first equation and substituting it
into the second equation).
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72 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How and when are systems of two linear equations solved precisely and graphically?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
HS.MP.7. Look for and make use of structure.
HS.MP.8. Look for and express regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Solve systems of linear equations by
any method.
Justify the method used to solve
systems of linear equations exactly
and approximately focusing on pairs
of linear equations in two variables.
EXPLANATIONS
AND EXAMPLES
The system solution methods can include but are not limited to graphical, elimination/linear combination,
substitution, and modeling. Systems can be written algebraically or can be represented in context. Students may use
graphing calculators, programs, or applets to model and find approximate solutions for systems of equations.
By now, students should know that a linear equation given by y = mx + b (or a variation of it) and makes a line on
a graph. They should also know that a system of linear equations means that we have more than one y = mx + b
equation in the mix. So far, so good.
Solving a system of linear equations means finding the point at which the two (or more?) lines intersect. This happens
when the same set of x and y values satisfy all the linear equations in the system.
What about lines that never intersect? Well, they’re parallel, for starters. But that just means that the system of parallel
lines will have no solution. (It may be helpful to tell your students that “No solution” is a legitimate answer, but only
after they’ve done the work to prove it. Otherwise you’ll get “No solution” as the answer to every homework problem.)
Students should be shown that a system of linear equations can be solved either through graphs or straight algebra,
but that these two methods arrive at the same answer because they mean the same thing. The goal is to find the point
at which the two lines intersect.
Graphs
Graphically, solving a system of two linear equations is easy. Using the slopes and y-intercepts of both lines, students
can graph each line individually and find the intersection point. It’s pretty and simple (and pretty simple), but not
always accurate visually. For instance, what’s the solution to the following system of equations?
STANDARD AND DECONSTRUCTION
A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
DESCRIPTION A.REI.6 Solve systems of equations using graphs.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 73
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
Is it at point (6.2, 17)? Or maybe (6.3, 17.5)? It’s difficult to tell exactly. Very helpful visually, but a bit less useful in
terms of coming up with numerical answers.
Algebra
It’d be helpful to know that the two lines above have equations of y = 2x + 5 and y = 3x – 1.25. With the linear
equations, students should be able to graph the lines on a coordinate plane as well as solve them exactly using
substitution. When the lines intersect, the x and y values in both linear equations are the same. That means we can
set the two equations equal to each other.
For instance, we can say that 3x – 1.25 = 2x + 5 because both the y coordinates have to be equal. (By the way, that
gives x = 6.25 as our answer. If we substitute 6.25 for x back into either equation, we should get 17.5.)
If students are struggling with these concepts, emphasizing the connection between the visual graphs and the
algebraic calculations should help cement their understanding..
(Source: www.shmoop.com)
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74 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• What is the most efficient method to solve systems of 2 equations that include a quadratic and linear equation,
algebraically and graphically?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
HS.MP.7. Look for and make use of structure.
HS.MP.8. Look for and express regularity in repeated reasoning.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Transform a simple system consisting
of a linear equation and a quadratic
equation in two variables so that a
solution can be found algebraically
and graphically.
Explain the correspondence between
the algebraic and graphical solutions
to a simple system consisting of
a linear equation and a quadratic
equation in two variables.
EXPLANATIONS
AND EXAMPLES
Students should know that this is a slight expansion on the previous standard. Our only change is that we instead of
having two lines, we have a linear equation (a straight line) and a quadratic equation (a not so straight line). No big
deal.
In the simplest terms, the quadratic equation is just a linear equation with a square sign over a variable. It can have
more terms, but as long as the largest exponent over a variable is 2, it’s a quadratic equation. Simple enough, right?
Students must know the quadratic formula:
Actually, they should have it tattooed on their faces. It’ll be a great icebreaker at parties.
The variables in the quadratic equation (lovingly named a, b, and c) are derived from our standard quadratic equation
ax2 + bx + c = 0.
Let’s use the actual problem in the standard itself as an example. We have y = -3x as our linear equation, and x2 + y2
= 3 as our quadratic. (Technically, the second equation isn’t a “true” quadratic, but there’s no shame in torturing your
students a little bit to teach them some more. We’re only kidding about the torturing part.)
If we graph these bad boys, we can already see that we have two points of intersection.
STANDARD AND DECONSTRUCTION
A.REI.7
Solve a simple system consisting of a linear equation and a quadratic
equation in two variables algebraically and graphically. For example, find
the points of intersection between the line y = –3x and the circle x² + y²= 3.
DESCRIPTION A.REI.7 Solve a system containing a linear equation and a quadratic equation in two variables (conic sections
possible) graphically and symbolically.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 75
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
We can prove why this is the case algebraically and calculate the exact points of intersection. First, we can solve for
our quadratic as we did with linear equations: by substituting one equation into another.
That turns x2 + y2 = 3 into x2 + (-3x)2 = 3. We can eventually get it into the ax2 + bx + c = 0 format (which ends up
being 10x2 – 3 = 0) so that we can use the quadratic formula. In our case, a = 10, b = 0, and c = -3.
If we plug in those values into the quadratic formula, we get
x ≈ ±0.55
Since we know what x is now, we can simply plug those values into our y = -3x equation to get our answers of
±1.64. That means our points of intersection are (-0.55, 1.64) and (0.55, -1.64). It matches up with the graph, so
we’re done.
Students should also make use of the discriminant in order to figure out whether the line and quadratic intersect
once, twice, or not at all. The discriminant is the part of the quadratic formula under the radical: b2 – 4ac.
If the value of the discriminant is less than 0, there are no points of intersection. If the answer is equal to 0, there is
one point of intersection (a tangent line). If the answer is greater than 0, there are two points of intersection.
The value of our discriminant is (0)2 – (4 × 10 × -3) = 120. That means there are two points of intersection. Using the
graph, the quadratic formula, and this little tidbit, we’ve triple checked our answer. Talk about thorough..
(Source: www.shmoop.com)
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76 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How does a matric equation provide a viable representation of a system of linear equations?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Write a system of linear equations in
vector variable form.
Write a system of linear equations as a
single matric equation.
Solve systems of equations.
EXPLANATIONS
AND EXAMPLES
Students should understand that a system of equations can be expressed as a matrix of coefficients multiplied by a
vector matrix of variables. Sometimes, it’s even better to do it that way.
Your students will probably curse you for stressing matrices time and time again, but just reassure them that it’s for
their own good. In order to work with matrices, your students should already understand what they are and how to
perform various functions with them. The same goes for linear equations and systems of linear equations.
Students should know that a matrix equation takes the form AX = B, where A represents the coefficient of our
variables, X represents our variables, and B represents the output to our equations.
In order to turn linear equations into matrices, we have to rearrange them all into the same format. Among the
preferred formats are ax + by + cz = d. The linear equations y = 2x + 5 and y = 3x – 2 become -2x + y = 5 and -3x + y = -2.
Painless, no?
Then we take each term in the equations and split them up into their proper matrix. Taking the equations from above,
we’ll have the A matrix (the one with all the coefficients) equaling
Since our only variables are x and y (and we put the coefficients into the A matrix in that order), our X matrix becomes:
The final matrix, B, is what’s on the other side of our equal signs. That means it turns into:
In matrix form, it looks like this:
This standard only indicates that students should know how to represent systems of linear equations in matrix
equation form, but not why doing so is useful (aside from organizational purposes). That is covered in the next
standard.
(Source: www.shmoop.com)
ESSENTIAL
QUESTION(S)
• How does a matric equation provide a viable representation of a system of linear equations?
MATHEMATICAL
PRACTICE(S)
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Solve a system of linear equations
using inverse matrices.
Solve a system of linear equations
with three or more variables using
technology.
Find the inverse of a matrix.
EXPLANATIONS
AND EXAMPLES
Students will perform multiplication, addition, subtraction, and scalar multiplication of matrices. They will use the
inverse of a matrix to solve a matrix equation. Students may use graphing calculators, programs, or applets to model
and find solutions for systems of equations.
This is where actual calculations take place. We’ve finally gotten to the fun stuff. Your students may think otherwise,
but we know the truth.
Students should already know how to form a matrix equation of AX = B from a system of linear equations and be
familiar with the concept of inverse matrices. They will use the inverse of a matrix in order to solve for the variables of
the matrix that is formed from given equations.
Students should know that if the determinant of a square matrix is zero (ad – bc = 0), there is no inverse to the matrix.
A matrix that takes the form.
has an inverse of
Assuming we’ve already translated our equations -2x + y = 5 and -3x + y = -2 into matrix equation form, we can find
the inverse matrix of the A matrix (the one with all the coefficients).
STANDARD AND DECONSTRUCTION
A.REI.8 (+) Represent a system of linear equations as a single matrix equation in a
vector variable.
DESCRIPTION A.REI.8 Write a system of linear equations as a single matrix equation.
STANDARD AND DECONSTRUCTION
A.REI.9
(+) Find the inverse of a matrix if it exists and use it to solve systems of
linear equations (using technology for matrices of dimension 3 x 3 or
greater).
DESCRIPTION
A.REI.9 Find the inverse of the coefficient matrix in the equation, if it exists. Use the inverse of the coefficient matrix
to solve the system. Use technology for matrices with dimensions 3 by 3 or greater.
A.REI.9 Find the dimension of matrices.
A.REI.9 Understand when matrices can be multiplied.
A.REI.9 Understand that matrix multiplication is not commutative.
A.REI.9 Understand the concept of an identity matrix.
A.REI.9 Understand why multiplication by the inverse of the coefficient matrix yields a solution to the system (if it
exists).
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 77
MATHEMATICS
ESSENTIAL
QUESTION(S)
• How does a matric equation provide a viable representation of a system of linear equations?
MATHEMATICAL
PRACTICE(S)
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
HS.MP.7. Look for and make use of structure.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Solve a system of linear equations
using inverse matrices.
Solve a system of linear equations
with three or more variables using
technology.
Find the inverse of a matrix.
EXPLANATIONS
AND EXAMPLES
Students will perform multiplication, addition, subtraction, and scalar multiplication of matrices. They will use the
inverse of a matrix to solve a matrix equation. Students may use graphing calculators, programs, or applets to model
and find solutions for systems of equations.
This is where actual calculations take place. We’ve finally gotten to the fun stuff. Your students may think otherwise,
but we know the truth.
Students should already know how to form a matrix equation of AX = B from a system of linear equations and be
familiar with the concept of inverse matrices. They will use the inverse of a matrix in order to solve for the variables of
the matrix that is formed from given equations.
Students should know that if the determinant of a square matrix is zero (ad – bc = 0), there is no inverse to the matrix.
A matrix that takes the form.
has an inverse of
Assuming we’ve already translated our equations -2x + y = 5 and -3x + y = -2 into matrix equation form, we can find
the inverse matrix of the A matrix (the one with all the coefficients).
STANDARD AND DECONSTRUCTION
A.REI.9
(+) Find the inverse of a matrix if it exists and use it to solve systems of
linear equations (using technology for matrices of dimension 3 x 3 or
greater).
DESCRIPTION
A.REI.9 Find the inverse of the coefficient matrix in the equation, if it exists. Use the inverse of the coefficient matrix
to solve the system. Use technology for matrices with dimensions 3 by 3 or greater.
A.REI.9 Find the dimension of matrices.
A.REI.9 Understand when matrices can be multiplied.
A.REI.9 Understand that matrix multiplication is not commutative.
A.REI.9 Understand the concept of an identity matrix.
A.REI.9 Understand why multiplication by the inverse of the coefficient matrix yields a solution to the system (if it
exists). RE
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78 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
First, we take our matrix and find its determinant. That would be -2 × 1 – 1 × -3 = -2 + 3 = 1. The inverse matrix
exists. Now, let’s give our creation life! After switching around the numbers, we have to multiply by the inverse of
the determinant, which is 1⁄1 or just 1. So our inverse matrix is fine as is.
So your students have found the inverse matrix. Now what? Well, the whole point of this process is to find the
values of x and y. (They should write that down. That’s important.) To do that, we can multiply the inverse matrix
we found by the B matrix (the one with all the solutions). That should give us the values for x and y.
That gives us 1 × 5 + (-1 × -2) = 5 + 2 = 7 = x, and 3 × 5 + (-2 × -2) = 15 + 4 = 19 = y. If we plug those values back
into the original linear equations, they should all hold up.
While this method might seem a bit cumbersome, students should appreciate that it’s especially useful for systems
of equations with many, many variables. On the other hand, the inverses for matrices larger than 2 × 2 should be
calculated using technology. The whole point, though, is that they can help us solve for the values of the variables.
(Source: www.shmoop.com)
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 79
MATHEMATICS
CLUSTER 4. Represent and solve equations and inequalities graphically.
BIG IDEA • Relationships between two sets of numbers can be described by mathematical rules, where a
function is a unique rule that has a one-to-one correspondence.
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80 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How does a graphed solution of equations and inequalities in two variables indicate the set of all its solutions?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Recognize that the graphical
representation of an equation in two
variables is a curve, which may be a
straight line.
Explain why each point on a curve is a
solution to its equation.
EXPLANATIONS
AND EXAMPLES
Students should understand that equations with two variables can be represented graphically. The shape that results
on the coordinate plane is a visual representation of all the solutions to that equation.
What does that mean? It means we’re not just pulling rabbits out of hats! The equations actually mean something
visually.
An equation with two variables can be anything from y = x to x2 + y2 = 4 to 19x13 = y. Some are simpler than others, of
course, but they all have an x and a y.
That means instead of having an equation with one variable (and therefore one solution), we can have many different
solutions. Graphically, we can represent these solutions by drawing a curve or line through all the pairs of solutions
(one for x and one for y) that work for that particular equation.
Let’s take the equation 7x – 18 = y and see how we can represent this graphically.
How do we prove to a student that indeed, a line of a two-variable equation, when graphed, shows all of the
solutions? Let’s show them how to pull that rabbit out of the hat themselves.
Since any two points define a line, all we need to do is input two values for x and see what the output y values are.
We’ll pick three points just to be sure our graph is a line and not some weird curve.
Let’s pick the numbers -1, 0, and 3 for x. Plugging in the numbers for x into our equation 7x – 18 = y gives us -25,
-18, and 3 for the y values. So the points in our graph become (-1, -25), (0, -18), and (3, 3). If we graph these on the
x-y coordinate plane, we’ll have this:
STANDARD AND DECONSTRUCTION
A.REI.10
Understand that the graph of an equation in two variables is the set of all
its solutions plotted in the coordinate plane, often forming a curve (which
could be a line).
DESCRIPTION A.REI.10 Understand that all solutions to an equation in two variables are contained on the graph of that equation.
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 81
MATHEMATICS
EXPLANATIONS
AND EXAMPLES
(continued)
If your students don’t believe you, prove to them that the equation and graph correspond to one another. Take
a point on the line that is easily identifiable, say (2, -4), and plug the values into the equation. If we do that, we’ll
have -4 = 7(2) – 18, which simplifies to -4 = -4. That way, students will be sure that points on the line or curve are
valid solutions to the equation, and vice versa.
But don’t stop there. It’s also important to prove the opposite. For example, the coordinate (4, 1), which is not on
the line, is also not a solution to our equation. If we plug in the coordinates, we can confirm this: 1 = 7(4) – 18 is
false because 1 ≠ 10. This means (4, 1) isn’t a solution to our equation and not a point on the line.
Now you can pat yourself on the back and prove to students that teachers aren’t just up to some magic tricks.
Everything in math pretty much works as it’s supposed to.
This method can be applied to two-variable equations of higher orders. The generic shapes of these equations
(such as quadratics making a parabola) should be known and associated with each other already. Otherwise,
students will need to graph several points before verifying the graph that corresponds with the particular
equation.
(Source: www.shmoop.com)
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82 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
ESSENTIAL
QUESTION(S)
• How does a graphed solution of equations and inequalities in two variables indicate the set of all its solutions?
MATHEMATICAL
PRACTICE(S)
HS.MP.2. Reason abstractly and quantitatively.
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
HS.MP.6. Attend to precision.
DOK Range Target
for Instruction &
Assessment
T 1 o 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and procedures. Tasks assessing expressing mathematical
reasoning.
Tasks assessing modeling/
applications.
Students should
be able to:
Recognize that if (x₁, y₁) and (x₂, y₂) share
the same location in the coordinate
plane that x₁ = x₂ and y₁ = y₂.
Recognize that f(x) = g(x) means that
there may be particular inputs of f and
g for which the outputs of f and g are
equal.
Recognize and use function notation to
represent linear, polynomial, rational,
absolute value, exponential, and
logarithmic equations.
Explain why the x-coordinates
of the points where the graph of
the equations y = f(x) and y = g(x)
intersect are the solutions of the
equations f(x) = g(x).
Approximate/find the solution(s)
using an appropriate method. For
example, using technology to graph
the functions, make tables of values
or find successive approximations.
EXPLANATIONS
AND EXAMPLES
Students need to understand that numerical solution methods (data in a table used to approximate an algebraic
function) and graphical solution methods may produce approximate solutions, and algebraic solution methods
produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators
or programs to generate tables of values, graph, or solve a variety of functions.
Students should understand that an equation and its graph are just two different representations of the same thing.
The graph of the line or curve of a two-variable equation shows in visual form all of the solutions (infinite as they may
be) to our equation in written form. When two equations are set to equal one another, their solution is the point at
which graphically they intersect one another. Depending on the equations (and the alignment of the planets), there
might be one solution, or more, or none at all.
Students can arrive at the correct answer(s) through graphing the functions and plotting their intersection points,
creating a table of x and f(x) values, and solving for x algebraically when f(x) = g(x). These strategies should be
provided to students and practiced with students so that the connection between graphs and equations are
solidified. (We wouldn’t want them to be liquefied, now would we?)
(Source: www.shmoop.com)
STANDARD AND DECONSTRUCTION
A.REI.11
Explain why the x-coordinates of the points where the graphs of the
equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x)
= g(x); find the solutions approximately, e.g., using technology to graph
the functions, make tables of values, or find successive approximations.
Include cases where f(x) and/or g(x) are linear, polynomial, rational,
absolute value, exponential, and logarithmic functions. («)
DESCRIPTION
A.REI.11 Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear,
polynomial, rational, absolute value, exponential, and logarithmic functions. Find the solution(s) by:
• Using technology to graph the equations and determine their point of intersection.
• Using tables of values.
• Using successive approximations that become closer and closer to the actual value.
ESSENTIAL
QUESTION(S)
• How does a graphed solution of equations and inequalities in two variables indicate the set of all its solutions?
MATHEMATICAL
PRACTICE(S)
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Identify characteristics of a linear
inequality and a system of linear
inequalities, such as: boundary
line, shading, and determine the
appropriate points to test and
derive a solution set from.
Explain the meaning of the
intersection of the shaded regions
in a system of linear inequalities.
Graph a line, or boundary line, and
shade the appropriate region for a
two variable linear inequality.
Graph a system of linear
inequalities and shade the
appropriate overlapping region
for a system of linear inequalities.
EXPLANATIONS
AND EXAMPLES
Students may use graphing calculators, programs, or applets to model and find solutions for inequalities or systems of
inequalities.
All this is asking us to do is what we already know from the previous standards, plus one simple step. In fact, this step
is fun (as long as you color inside the lines). Students should know how to graph a linear inequality, complete with all
the nuts and bolts.
A linear inequality is the same as a linear equation, but instead of an equal sign, we’ll have to use the inequality signs
(like ≤, ≥, <, and >).
What’s all this “half-plane” business? Just mathematical mumbo-jumbo. It means that because we’re graphing an
inequality and our linear equation is with a different sign now, it’ll be shaded above or below the line as part of our
solution. That’s it.
When dealing with inequalities, your students should ask themselves two questions:
1. Which part of the graph do I shade in?
2. Do I draw a dotted or a solid line?
If the inequality is greater than, or greater than or equal to (using either > or ≥), then we shade the upper half of the
graph. If the inequality if less than, or less than or equal to (using either < or ≤), then we shade the lower half of the
graph.
If students are struggling with which half to shade, the simplest way to remove all doubt is to plug in the coordinates
of a point that’s very obviously on one side of the boundary. If the inequality is true for that point, then we know to
shade the “half-plane” containing that point. If it’s false, we’ll shade in the other half. Make sure to bring your colored
pencils.
STANDARD AND DECONSTRUCTION
A.REI.12
Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph
the solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
DESCRIPTION
A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary for non-
inclusive inequalities.
A.REI.12 Graph the solution set to a system of linear inequalities in two variables as the intersection of their
corresponding half-planes.
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 83
MATHEMATICS
ESSENTIAL
QUESTION(S)
• How does a graphed solution of equations and inequalities in two variables indicate the set of all its solutions?
MATHEMATICAL
PRACTICE(S)
HS.MP.4. Model with mathematics.
HS.MP.5. Use appropriate tools strategically.
DOK Range Target
for Instruction &
Assessment
T 1 T 2 o 3 o 4
Learning Expectations Know: Concepts/Skills Think Do
Assessment Types Tasks assessing concepts, skills, and
procedures.
Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.
Students should
be able to:
Identify characteristics of a linear
inequality and a system of linear
inequalities, such as: boundary
line, shading, and determine the
appropriate points to test and
derive a solution set from.
Explain the meaning of the
intersection of the shaded regions
in a system of linear inequalities.
Graph a line, or boundary line, and
shade the appropriate region for a
two variable linear inequality.
Graph a system of linear
inequalities and shade the
appropriate overlapping region
for a system of linear inequalities.
EXPLANATIONS
AND EXAMPLES
Students may use graphing calculators, programs, or applets to model and find solutions for inequalities or systems of
inequalities.
All this is asking us to do is what we already know from the previous standards, plus one simple step. In fact, this step
is fun (as long as you color inside the lines). Students should know how to graph a linear inequality, complete with all
the nuts and bolts.
A linear inequality is the same as a linear equation, but instead of an equal sign, we’ll have to use the inequality signs
(like ≤, ≥, <, and >).
What’s all this “half-plane” business? Just mathematical mumbo-jumbo. It means that because we’re graphing an
inequality and our linear equation is with a different sign now, it’ll be shaded above or below the line as part of our
solution. That’s it.
When dealing with inequalities, your students should ask themselves two questions:
1. Which part of the graph do I shade in?
2. Do I draw a dotted or a solid line?
If the inequality is greater than, or greater than or equal to (using either > or ≥), then we shade the upper half of the
graph. If the inequality if less than, or less than or equal to (using either < or ≤), then we shade the lower half of the
graph.
If students are struggling with which half to shade, the simplest way to remove all doubt is to plug in the coordinates
of a point that’s very obviously on one side of the boundary. If the inequality is true for that point, then we know to
shade the “half-plane” containing that point. If it’s false, we’ll shade in the other half. Make sure to bring your colored
pencils.
STANDARD AND DECONSTRUCTION
A.REI.12
Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph
the solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
DESCRIPTION
A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary for non-
inclusive inequalities.
A.REI.12 Graph the solution set to a system of linear inequalities in two variables as the intersection of their
corresponding half-planes.
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84 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT
HIGH SCHOOL ALGEBRA
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,
11TH – 12TH GRADES: 1185L TO 1385L
EXPLANATIONS
AND EXAMPLES
(continued)
The line that graphs our linear equation is dashed or dotted if we use greater than or less than (using > or <) in
our inequality. That’s so we know the line is a boundary, but all the points on it don’t satisfy the inequality. The
line we’ll use is solid if the inequality has a greater than or equal to, or less than or equal to (using ≥ or ≤), symbol
because the boundary includes possible solutions to our inequality.
Students should understand how to graph not one, but two inequalities. It’s just like graphing one inequality, and
then graphing another right on top of it. Using the same graph saves trees.
Given a pair of inequalities (such as y < x – 5 and y ≥ x – 6, for instance), we draw them as though they were
equations first. We can do this through a computer, a graphing calculator, or by creating a table of values to
calculate enough points to get us a straight line.
Time to bust out those colored pencils. Since our first inequality is “less than,” this means we must shade below the
line. We’ll color it red. Then comes the ultimate question: solid or dotted? Well, there’s no “equal to” component, so
our set of solutions to the inequality does not include the boundary line itself. That means it must be drawn as a
dotted line.
For the second inequality, we know that it must be “greater than or equal to,” meaning we shade above the line.
We’ll pick blue. Because of its “equal to” part, we must include the line. It must remain solid.
Red and blue make purple. The overlapping purple area is the solution to our system of inequalities. That means
that only within the overlapping area will the values of x and y work for both the inequalities we listed.
Students should know how to graph inequalities, shade in the half-planes, and find the set of solutions for a
system of inequalities. If students are struggling, have them plug in coordinates that are on the boundary or very
clearly to one side. This will help connect the graph and the inequality, as well as make sense of what’s going
algebraically and graphically.
Also, make sure they pick colors that go together.
(Source: www.shmoop.com)
http://www.shmoop.com
COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 85
MATHEMATICS
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855.809.7018 | www.commoncoreinstitute.com
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