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High School
CCSS
Mathematics II
Curriculum
Guide
-Quarter 3-
Columbus City
Schools
Page 1 of 144
Table of Contents
Math Practices Rationale .............................................................................................................................................. 3
RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE .................................................................. 11
Mathematical Practices: A Walk-Through Protocol .................................................................................................... 16
Curriculum Timeline .................................................................................................................................................... 19
Scope and Sequence.................................................................................................................................................... 20
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 ....................... 30
Teacher Notes .......................................................................................................................................................... 32
Families of Graphs #2 .......................................................................................................................................... 58
Solving By Factoring ............................................................................................................................................ 64
Polynomial Cards ................................................................................................................................................. 66
Finding the Greatest Common Monomial Using Algebra Tiles ........................................................................... 70
Polynomial Cards ................................................................................................................................................. 77
Drawkcab Problems............................................................................................................................................. 80
Discovering the Difference of Two Squares ........................................................................................................ 88
Factoring Using the Greatest Common Factor .................................................................................................... 92
Factoring By Grouping ......................................................................................................................................... 94
Factoring Worksheet ........................................................................................................................................... 96
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8,
8a; A – CED 1; N – CN 1, 2, 7 ...................................................................................................................................... 100
Teacher Notes:....................................................................................................................................................... 102
Sorting Activity .................................................................................................................................................. 118
Learning How to Complete the Square “Completely” ...................................................................................... 122
Transformations and Completing the Square Notes ......................................................................................... 126
Completing the Square and Transformations Practice ..................................................................................... 134
Discovery of Completing the Square ................................................................................................................. 138
Page 2 of 144
Math Practices Rationale
CCSSM Practice 1: Make sense of problems and persevere in solving them.

Why is this practice important?


What does this practice look like when students are
doing it?


What can a teacher do to model this practice?


What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?



What actions might the teacher make that inhibit
the students’ use of this practice?



Helps students to develop critical thinking
skills.
Teaches students to “think for themselves”.
Helps students to see there are multiple
approaches to solving a problem.
Students immediately begin looking for
methods to solve a problem based on previous
knowledge instead of waiting for teacher to
show them the process/algorithm.
Students can explain what problem is asking as
well as explain, using correct mathematical
terms, the process used to solve the problem.
Frame mathematical questions/challenges so
they are clear and explicit.
Check with students repeatedly to help them
clarify their thinking and processes.
“How would you go about solving this
problem?”
“What do you need to know in order to solve
this problem?”
What methods have we studied that you can
use to find the information you need?
Students can explain the relationships
between equations, verbal descriptions,
tables, and graphs.
Students check their answer using a different
method and continually ask themselves, “Does
this make sense?”
They understand others approaches to solving
complex problems and can see the similarities
between different approaches.
Showing the students shortcuts/tricks to solve
problems (without making sure the students
understand why they work).
Not giving students an adequate amount of
think time to come up with solutions or
processes to solve a problem.
Giving students the answer to their questions
instead of asking guiding questions to lead
them to the discovery of their own question.
Page 3 of 144
CCSSM Practice 2: Reason abstractly and quantitatively.

Why is this practice important?

What does this practice look like when students are
doing it?



What can a teacher do to model this practice?



What questions could a teacher ask to encourage
the use of this practice?



Students develop reasoning skills that help
them to understand if their answers make
sense and if they need to adjust the answer to
a different format (i.e. rounding)
Students develop different ways of seeing a
problem and methods of solving it.
Students are able to translate a problem
situation into a number sentence or algebraic
expression.
Students can use symbols to represent
problems.
Students can visualize what a problem is
asking.
Ask students questions about the types of
answers they should get.
Use appropriate terminology when discussing
types of numbers/answers.
Provide story problems and real world
problems for students to solve.
Monitor the thinking of students.
“What is your unknown in this problem?
“What patterns do you see in this problem and
how might that help you to solve it?”

What does proficiency look like in this practice?


What actions might the teacher make that inhibit
the students’ use of this practice?

Students can recognize the connections
between the elements in their mathematical
sentence/expression and the original problem.
Students can explain what their answer
means, as well as how they arrived at it.
Giving students the equation for a word or
visual problem instead of letting them “figure
it out” on their own.
Page 4 of 144
CCSSM Practice 3: Construct viable arguments and critique the reasoning of others

Why is this practice important?

What does this practice look like when students are
doing it?


What can a teacher do to model this practice?






What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?



What actions might the teacher make that inhibit
the students’ use of this practice?


Students better understand and remember
concepts when they can defend and explain
it to others.
Students are better able to apply the
concept to other situations when they
understand how it works.
Communicate and justify their solutions
Listen to the reasoning of others and ask
clarifying questions.
Compare two arguments or solutions
Question the reasoning of other students
Explain flaws in arguments
Provide an environment that encourages
discussion and risk taking.
Listen to students and question the clarity of
arguments.
Model effective questioning and appropriate
ways to discuss and critique a mathematical
statement.
How could you prove this is always true?
What parts of “Johnny’s “ solution confuses
you?
Can you think of an example to disprove
your classmates theory?
Students are able to make a mathematical
statement and justify it.
Students can listen, critique and compare
the mathematical arguments of others.
Students can analyze answers to problems
by determining what answers make sense.
Explain flaws in arguments of others.
Not listening to students justify their
solutions or giving adequate time to critique
flaws in their thinking or reasoning.
Page 5 of 144
CCSSM Practice 4: Model with mathematics
Why is this practice important?

Helps students to see the connections
between math symbols and real world
problems.
What does this practice look like when students are
doing it?


Write equations to go with a story problem.
Apply math concepts to real world problems.
What can a teacher do to model this practice?

Use problems that occur in everyday life and
have students apply mathematics to create
solutions.
Connect the equation that matches the real
world problem. Have students explain what
different numbers and variables represent in
the problem situation.
Require students to make sense of the
problems and determine if the solution is
reasonable.
How could you represent what the problem
was asking?
How does your equation relate to the
problems?
How does your strategy help you to solve
the problem?
Students can write an equation to represent
a problem.
Students can analyze their solutions and
determine if their answer makes sense.
Students can use assumptions and
approximations to simplify complex
situations.
Not give students any problem with real
world applications.


What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?



What actions might the teacher make that inhibit
the students’ use of this practice?

Page 6 of 144
CCSSM Practice 5: Use appropriate tools strategically
Why is this practice important?

What does this practice look like when students are
doing it?



What can a teacher do to model this practice?



What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?


What actions might the teacher make that inhibit
the students’ use of this practice?


Helps students to understand the uses and
limitations of different mathematical and
technological tools as well as which ones can
be applied to different problem situations.
Students select from a variety of tools that
are available without being told which to
use.
Students know which tools are helpful and
which are not.
Students understand the effects and
limitations of chosen tools.
Provide students with a variety of tools
Facilitate discussion regarding the
appropriateness of different tools.
Allow students to decide which tools they
will use.
How is this tool helping you to understand
and solve the problem?
What tools have we used that might help
you organize the information given in this
problem?
Is there a different tool that could be used to
help you solve the problem?
Students are sufficiently familiar with tools
appropriate for their grade or course and
make sound decisions about when each of
these tools might be helpful.
Students recognize both the insight to be
gained from the use of the selected tool and
their limitations.
Only allowing students to solve the problem
using one method.
Telling students that the solution is incorrect
because it was not solved “the way I showed
you”.
Page 7 of 144
CCSSM Practice 6: Attend to precision.

Why is this practice important?

What does this practice look like when students are
doing it?


What can a teacher do to model this practice?



What questions could a teacher ask to encourage
the use of this practice?

What does proficiency look like in this practice?





What actions might the teacher make that inhibit
the students’ use of this practice?



Students are better able to understand new
math concepts when they are familiar with
the terminology that is being used.
Students can understand how to solve real
world problems.
Students can express themselves to the
teacher and to each other using the correct
math vocabulary.
Students use correct labels with word
problems.
Make sure to use correct vocabulary terms
when speaking with students.
Ask students to provide a label when
describing word problems.
Encourage discussions and explanations and
use probing questions.
How could you describe this problem in your
own words?
What are some non-examples of this word?
What mathematical term could be used to
describe this process.
Students are precise in their descriptions.
They use mathematical definitions in their
reasoning and in discussions.
They state the meaning of symbols
consistently and appropriately.
Teaching students “trick names” for symbols
(i.e. the alligator eats the big number)
Not using proper terminology in the
classroom.
Allowing students to use the word “it” to
describe symbols or other concepts.
Page 8 of 144
CCSSM Practice 7: Look for and make use of structure.
Why is this practice important?

When students can see patterns or
connections, they are more easily able to
solve problems
What does this practice look like when students are
doing it?

Students look for connections between
properties.
Students look for patterns in numbers,
operations, attributes of figures, etc.
Students apply a variety of strategies to
solve the same problem.
Ask students to explain or show how they
solved a problem.
Ask students to describe how one repeated
operation relates to another (addition vs.
multiplication).
How could you solve the problem using a
different operation?
What pattern do you notice?


What can a teacher do to model this practice?


What questions could a teacher ask to encourage
the use of this practice?

What does proficiency look like in this practice?

Students look closely to discern a pattern or
structure.
What actions might the teacher make that inhibit
the students’ use of this practice?

Provide students with pattern before
allowing them to discern it for themselves.

Page 9 of 144
CCSSM Practice 8: Look for and express regularity in repeated reasoning
Why is this practice important?

What does this practice look like when students are
doing it?

What can a teacher do to model this practice?




What questions could a teacher ask to encourage
the use of this practice?




When students discover connections or
algorithms on their own, they better
understand why they work and are more
likely to remember and be able to apply
them.
Students discover connections between
procedures and concepts
Students discover rules on their own
through repeated exposures of a concept.
Provide real world problems for students to
discover rules and procedures through
repeated exposure.
Design lessons for students to make
connections.
Allow time for students to discover the
concepts behind rules and procedures.
Pose a variety of similar type problems.
How would you describe your method? Why
does it work?
Does this method work all the time?
What do you notice when…?
What does proficiency look like in this practice?


Students notice repeated calculations.
Students look for general methods and
shortcuts.
What actions might the teacher make that inhibit
the students’ use of this practice?

Providing students with formulas or
algorithms instead of allowing them to
discover it on their own.
Not allowing students enough time to
discover patterns.

Page 10 of 144
RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE
Using the Rubric:
Review each row corresponding to a mathematical practice. Use the boxes to mark the appropriate description for your task or teacher action. The
task descriptors can be used primarily as you develop your lesson to make sure your classroom tasks help cultivate the mathematical practices. The
teacher descriptors, however, can be used during or after the lesson to evaluate how the task was carried out. The column titled “proficient”
describes the expected norm for task and teacher action while the column titled “exemplary” includes all features of the proficient column and more.
A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.
PRACTICE
NEEDS IMPROVEMENT
Task:
Make sense of
problems and
persevere in
solving them.


EMERGING
PROFICIENT
EXEMPLARY
(teacher does thinking)
(teacher mostly models)
(students take ownership)
Task:
Is strictly procedural.
Does not require students
to check solutions for
errors.


Task:
Is overly scaffolded or
procedurally “obvious”.
Requires students to
check answers by plugging
in numbers.


Allots too much or too
little time to complete
task.
Encourages students to
individually complete
tasks, but does not ask
them to evaluate the
processes used.
Explains the reasons
behind procedural steps.
Does not check errors
publicly.


Teacher:



Does not allow for wait
time; asks leading
questions to rush through
task.
Does not encourage
students to individually
process the tasks.
Is focused solely on
answers rather than
processes and reasoning.
Teacher:








Allows ample time for all
students to struggle with
task.
Expects students to
evaluate processes
implicitly.
Models making sense of
the task (given situation)
and the proposed
solution.
Allows for multiple entry
points and solution paths.
Requires students to
defend and justify their
solution by comparing
multiple solution paths.
Teacher:


Teacher:

Page 11 of 144
Task:
Is cognitively demanding.
Has more than one entry
point.
Requires a balance of
procedural fluency and
conceptual
understanding.
Requires students to
check solutions for errors
using one other solution
path.

Differentiates to keep
advanced students
challenged during work
time.
Integrates time for explicit
meta-cognition.
Expects students to make
sense of the task and the
proposed solution.
PRACTICE
Reason
abstractly and
quantitatively.
EMERGING
NEEDS IMPROVEMENT


Lacks context.
Does not make use of
multiple representations
or
solution paths.

 Is embedded in a
contrived context.


Teacher:

 Expects students to


Teacher:

(teacher does thinking)
Task:

Task:
Does not expect
students to interpret
representations.
Expects students to
memorize procedures
withno connection to
meaning.





model and interpret
tasks using a single
representation.
Explains connections
between procedures and
meaning.








Task:

Task:
Construct viable
arguments and
critique the
reasoning of
others.

Is either ambiguously
stated.
Teacher:






EXEMPLARY
(teacher mostly models)
(students take ownership)
Task:

Task:

 Has realistic context.
 Has relevant realistic

Requires
students
to
context.


frame solutions in a


context.
Teacher:


 Has solutions that can be
 Expects students to


expressed with multiple
interpret, model, and


representations.
connect multiple


representations.

 Prompts students to
Teacher:
 Expects students to

articulate connections


interpret and model
between algebraic


using multiple
procedures and contextual


representations.
meaning.
 Provides structure for



students to connect
algebraic
procedures
to


contextual meaning.


 Links mathematical

solution with a


question’s answer.


Task:

Is not at the appropriate
level.
Teacher:
Does not ask students to
present arguments or
solutions.
Expects students to
follow a given solution
path without
opportunities to
make conjectures.
PROFICIENT

Teacher:

Avoids single steps or
routine algorithms.

Identifies students’
assumptions.
Models evaluation of
student arguments.
Asks students to explain
their conjectures.

Teacher:
Does not help students
differentiate between
assumptions and logical
conjectures.
Asks students to present
arguments but not to
evaluate them.
Allows students to make
conjectures without
justification.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Visualizing Functions




Helps students
differentiate between
assumptions and logical
conjectures.
Prompts students to
evaluate peer arguments.
Expects students to
formally justify the validity
of their conjectures.
Summer 2011
Page 12 of 144
PRACTICE
Task:
Model with
mathematics.
EMERGING
NEEDS IMPROVEMENT

(teacher does thinking)
Task:
Requires students to

Requires students to
identify variables and to
perform necessary
computations.
Teacher:

Identifies appropriate
Does not discuss

Requires students to
identify variables and to
compute and interpret


report findings using a
mixture of
representations.
Verifies that students have
identified appropriate
variables and procedures.
appropriateness of model.

Illustrates the relevance of
the mathematics involved.

Explains the
EXEMPLARY
(students take ownership)
Task:

Requires students to
identify variables, compute
and interpret results, and
results.
Teacher:
variables and procedures
for students.

PROFICIENT
(teacher mostly models)
Task:
Requires students to
appropriateness of model.
identify variables, compute
and interpret results,
report findings, and justify
the reasonableness of their
results and procedures
within context of the task.
Teacher:

Expects students to justify
identify extraneous or
missing information.
Teacher:

their choice of variables
and procedures.

Gives students opportunity
to evaluate the
appropriateness of model.
Asks questions to help
students identify
appropriate variables and
procedures.

Facilitates discussions in
evaluating the
appropriateness of model.
Use appropriate
tools strategically.
Task:

Task:
Does not incorporate
additional learning tools.
Teacher:


Task:
Lends itself to one learning

tool.

additional learning tools.
Does not involve mental
Teacher:


learning tools.

computations or
estimation.
Does not incorporate
Task:
Lends itself to multiple
Gives students opportunity
to develop fluency in
mental computations.

Demonstrates use of
appropriate learning tool.
Chooses appropriate
learning tools for student
use.

Models error checking by
estimation.
tools (i.e., graph paper,
calculator, manipulative).
Requires students to
demonstrate fluency in
Teacher:

Requires multiple learning
mental computations.
Teacher:

Allows students to choose
appropriate learning tools.

Creatively finds
appropriate alternatives
where tools are not
available.
Page 13 of 144
PRACTICE
Attend to
precision.
EMERGING
NEEDS IMPROVEMENT

(teacher does thinking)
Task:

Task:
Gives imprecise
instructions.


Has overly detailed or
wordy instructions.

Teacher:


Look for and make
use of structure.
Teacher:

Does not intervene
when students are being
imprecise.
Does not point out
instances when students
fail to address the
question completely or
directly.
Requires students to
automatically apply an
algorithm to a task
without evaluating its
appropriateness.








Teacher:



Teacher:



Inconsistently intervenes
when students are
imprecise.
Identifies incomplete
responses but does not
require student to
formulate further
response.
Task:

Task:





Does not recognize
students for developing
efficient approaches to
the task.
Requires students to
apply the same
algorithm to a task
although there may be
other approaches.










PROFICIENT
(teacher mostly models)
(students take ownership)
Task:

Task:

 Has precise instructions.
 Includes assessment
Teacher:


criteria for communication
of ideas.
 Consistently demands

precision in
Teacher:


communication and in

 Demands and models
mathematical solutions.
precision in



Identifies
incomplete
communication and in


responses
and
asks
mathematical solutions.


student
to
revise
their
 Encourages students to


response.
identify when others are


not addressing the


question completely.




Task:

Requires students to
analyze a task before
automatically applying
an algorithm.
Identifies individual
students’ efficient
approaches, but does
not expand
understanding to
the rest of the class.
Demonstrates the same
algorithm to all related
tasks although there
may be other more
effective
approaches.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Visualizing Functions
EXEMPLARY





Task:

Requires students to
analyze a task and
identify more than one
approach
to the problem.










Facilitates all students in
developing reasonable
and
efficient ways to
accurately perform basic
operations.
Continuously questions
students about the
reasonableness of their
intermediate results.
Requires students to
identify the most efficient
solution to the task.

Teacher:




Teacher:














Prompts students to
identify mathematical
structure of the task in
order to identify the most
effective solution path.
Encourages students to
justify their choice of
algorithm or solution path.

Summer 2011
Page 14 of 144
PRACTICE
Look for and
express regularity
in repeated
reasoning.
EMERGING
NEEDS IMPROVEMENT


Is disconnected from
prior and future
concepts.
Has no logical
progression that leads to
pattern recognition.



Does not show evidence
of understanding the
hierarchy within
concepts.
Presents or examines
task in isolation.
Is overly repetitive or
has gaps that do not
allow for development
of a pattern.


Teacher:

 Hides or does not draw

connections to prior or

Teacher:

(teacher does thinking)
Task:

Task:
future concepts.
PROFICIENT
EXEMPLARY
(teacher mostly models)
(students take ownership)
Task:

Task:

 Reviews prior knowledge
 Addresses and connects to
and requires cumulative
prior knowledge in a non

understanding.
routine way.
 Lends itself to

 Requires recognition of


developing a
pattern or structure to be


pattern or structure.
completed.



Teacher:












Teacher:
Connects concept to
prior and future
concepts to help
students develop an
understanding of
procedural shortcuts.
Demonstrates
connections between
tasks.



























Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Visualizing Functions

Encourages students to
connect task to prior
concepts and tasks.
Prompts students to
generate exploratory
questions based on the
current task.
Encourages students to
monitor each other’s
intermediate results.
Summer 2011
Page 15 of 144
Mathematical Practices: A Walk-Through Protocol
*Note: This document should also be used by the teacher for planning and self-evaluation.
Mathematical Practices
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.
Observations
Students are expected to______________:
 Engage in solving problems.
 Explain the meaning of a problem and restate in it their own words.
 Analyze given information to develop possible strategies for solving the problem.
 Identify and execute appropriate strategies to solve the problem.
 Check their answers using a different method, and continually ask “Does this make sense?”
Teachers are expected to______________:
 Provide time for students to discuss problem solving.
Students are expected to______________:
 Connect quantity to numbers and symbols (decontextualize the problem) and
create a logical representation of the problem at hand.
 Recognize that a number represents a specific quantity (contextualize the problem).
 Contextualize and decontextualize within the process of solving a problem.
Teachers are expected to______________:
 Provide appropriate representations of problems.
Students are expected to____________________________:
 Explain their thinking to others and respond to others’ thinking.
 Participate in mathematical discussions involving questions like “How did you get that?” and
“Why is that true?”
 Construct arguments that utilize prior learning.
 Question and problem pose.
 Practice questioning strategies used to generate information.
 Analyze alternative approaches suggested by others and select better approaches.
 Justify conclusions, communicate them to others, and respond to the arguments of others.
 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.
Teachers are expected to______________:
 Provide opportunities for students to listen to or read the conclusions and arguments
of others.
CCSSM
National Professional Development
Page 16 of 144
Mathematical Practices
MP.4. Model with mathematics.
MP 5. Use appropriate
tools strategically
Observations
Students are expected to______________:
 Apply the mathematics they know to solve problems arising in everyday life, society,
and the workplace.
 Make assumptions and approximations to simplify a complicated situation, realizing
that these may need revision later.
 Experiment with representing problem situations in multiple ways, including numbers,
words (mathematical language), drawing pictures, using objects, acting out, making a
chart or list, creating equations, etc.
 Identify important quantities in a practical situation and map their relationships using
such tools as diagrams, two-way tables, graphs, flowcharts, and formulas.
 Evaluate their results in the context of the situation and reflect on whether their results
make sense.
 Analyze mathematical relationships to draw conclusions.
Teachers are expected to______________:
 Provide contexts for students to apply the mathematics learned.
Students are expected to______________:
 Use tools when solving a mathematical problem and to deepen their understanding of
concepts (e.g., pencil and paper, physical models, geometric construction and measurement
devices, graph paper, calculators, computer-based algebra or geometry systems.)
 Consider available tools when solving a mathematical problem and decide when
certain tools might be helpful, recognizing both the insight to be gained and their
limitations.
 Detect possible errors by strategically using estimation and other mathematical knowledge.
Teachers are expected to______________:
CCSSM
National Professional Development
Page 17 of 144
Mathematical Practices
MP.6. Attend to precision.
MP.7. Look for and make use of
structure.
MP.8. Look for and express
regularity in repeated
reasoning.
Observations
Students are expected to______________:
 Use clear and precise language in their discussions with others and in their own reasoning.
 Use clear definitions and state the meaning of the symbols they choose, including using the
equal sign consistently and appropriately.
 Specify units of measure and label parts of graphs and charts.
 Calculate with accuracy and efficiency based on a problem’s expectation.
Teachers are expected to______________:
 Emphasize the importance of precise communication.
Students are expected to______________:
 Describe a pattern or structure.
 Look for, develop, generalize, and describe a pattern orally, symbolically, graphically and in
written form.
 Relate numerical patterns to a rule or graphical representation
 Apply and discuss properties.
Teachers are expected to______________:
 Provide time for applying and discussing properties.
Students are expected to______________:
 Describe repetitive actions in computation
 Look for mathematically sound shortcuts.
 Use repeated applications to generalize properties.
 Use models to explain calculations and describe how algorithms work.
 Use models to examine patterns and generate their own algorithms.
 Check the reasonableness of their results.
Teachers are expected to______________:
CCSSM
National Professional Development
Page 18 of 144
High School Common Core Math II
Curriculum Timeline
Topic
Intro Unit
Similarity
Trigonometric
Ratios
Other Types of
Functions
Comparing
Functions and
Different
Representations
of Quadratic
Functions
Modeling Unit
and Project
Quadratic
Functions: Solving
by Factoring
Quadratic
Functions:
Completing the
Square and the
Quadratic
Formula
Probability
Geometric
Measurement
Geometric
Modeling Unit
and Project
Standards Covered
G – SRT 1
G – SRT 1a
G – SRT 1b
G – SRT 6
G – SRT 2
G – SRT 3
G – SRT 4
G – SRT 7
G – SRT 5
Grading
Period
1
1
No. of
Days
5
20
G – SRT 8
1
20
A – CED 1
A – CED 4
A – REI 1
N – RN 1
N – RN 2
N – RN 3
F – IF 4
F – IF 5
F – IF 6
F – IF 7
F – IF 7a
F– IF 9
F – IF 4
F – IF 7b
F – IF 7e
F – IF 8
F – IF 8b
F– BF1
A– CED 1
A– CED 2
F– BF 1
F– BF 1a
F – BF 1b
F– BF 3
F – BF 1a
F – BF 1b
F – BF 3
A – SSE 1b
N–Q2
2
15
F – LE 3
N– Q 2
S – ID 6a
S – ID 6b
A – REI 7
2
20
2
10
A – APR 1
A – REI 1
A – REI 4b
F – IF 8a
A – CED 1
A – SSE 1b
A – SSE 3a
3
20
A – REI 1
A – REI 4
A – REI 4a
A – REI 4b
A – SSE 3b
F – IF 8
F – IF 8a
A – CED 1
N – CN 1
N – CN 2
N – CN 7
3
20
S – CP 1
S – CP 2
S – CP 3
G – GMD 1
S – CP 4
S – CP 5
S – CP 6
G – GMD 3
S – CP 7
4
20
4
10
G – MG 1
G – MG 2
G – MG 3
4
15
Page 19 of 144
High School Common Core Math II
1st Nine Weeks
Scope and Sequence
Intro Unit – IO (5 days)
Topic 1 – Similarity (20 days)
Geometry (G – SRT):
1) Similarity, Right Triangles, and Trigonometry:
Understand similarity in terms of similarity transformations.
 G – SRT 1: Verify experimentally the properties of dilations given by a center and a
scale factor.

G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.

G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.

G – SRT 2: Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity transformations the
meaning of similarity for triangles as the equality of all corresponding pairs of angles and
the proportionality of all corresponding pairs of sides.

G – SRT 3: Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Geometry (G – SRT):
2) Similarity, Right Triangles, and Trigonometry:
Prove theorems involving similarity.
 G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one
side of a triangle divides the other two proportionally, and conversely; the Pythagorean
Theorem proved using triangle similarity.

G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Topic 2 – Trigonometric Ratios (20 days)
Geometry (G – SRT):
3) Similarity, Right Triangles, and Trigonometry:
Define trigonometric ratios and solve problems involving .right triangles
 G – SRT 6: Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G – SRT 7: Explain and use the relationship between the sine and cosine of
complementary angles.

G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems.
Page 20 of 144
High School Common Core Math II
2nd Nine Weeks
Scope and Sequence
Topic 3 – Other Types of Functions (15 days)
Creating Equations (A – CED):
4) Create equations that describe numbers or relationships
 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.
 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.
Reasoning with Equations and Inequalities (A – REI):
5) Understand solving equations as a process of reasoning and explain the reasoning.
 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.
The Real Number System (N – RN):
6) Extend the properties of exponents to rational exponents.
 N – RN 1: Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a notation for
radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of
5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
 N – RN 2: Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
The Real Number System (N – RN):
7) Use properties of rational and irrational numbers.
 N – RN 3: Explain why the sum or product of two rational numbers is rational; that the
sum of a rational number and an irrational number is irrational; and that the product of a
nonzero rational number and an irrational number is irrational.
Interpreting Functions (F – IF):
8) Interpret functions that arise in applications in terms of the context.
 F – IF 4*: For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.*
Interpreting Functions (F – IF):
9) Analyze functions using different representations.
 F – IF 7b: Graph square root, cube root, and absolute value functions.

F – IF 7e: Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and amplitude.
Page 21 of 144

F – IF 8: Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.

F – IF 8b: Use the properties of exponents to interpret expressions for exponential
functions. For example, identify percent rate of change in functions such as y = (1.02)t, y
= (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth
or decay.
Building Functions (F – BF):
10) Build a function that models a relationship between two quantities.
 F – BF 1: Write a function that describes a relationship between two quantities.

F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation
from a context.

F – BF 1b: Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a constant
function to a decaying exponential, and relate these functions to the model.
Building Functions (F – BF):
11) Build new functions from existing functions.
 F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x
+ k) for specific values of k (both positive and negative); find the value of k given the
graphs. Experiment with cases and illustrate an explanation of the effects on the graph
using technology. Include recognizing even and odd functions from their graphs and
algebraic expressions for them.
Seeing Structure in Expressions (A – SSE):
12) Interpret the structure of expressions.
 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.
Quantities (NQ):
13) Reason quantitatively and use units to solve problems.
 N – Q 2: Define appropriate quantities for the purpose of descriptive modeling.
Topic 4 – Comparing Functions and Different Representations of Quadratic Functions (20
days)
Interpreting Functions (F – IF):
14) Interpret functions that arise in applications in terms of the context.
 F – IF 4*: For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.*

F – IF 5*: Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the number
Page 22 of 144
of person-hours it takes to assemble n engines in a factory, then the positive integers
would be an appropriate domain for the function.*

F – IF 6: Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of change from a
graph.
Interpreting Functions (F – IF):
15) Analyze functions using different representations.
 F – IF 7: Graph functions expressed symbolically and show key features of the graph, by
hand in simple cases and using technology for more complicated cases.

F – IF 7a*: Graph linear and quadratic functions and show intercepts, maxima, and
minima.*

F – IF 9: Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for
another, say which has the larger maximum.
Creating Equations (A – CED):
16) Create equations that describe numbers or relationships.
 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.

A – CED 2: Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels and scales.
Building Functions (F – BF):
17) Build a function that models a relationship between two quantities.
 F – BF 1: Write a function that describes a relationship between two quantities.

F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation
from a context.

F – BF 1b: Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a constant
function to a decaying exponential, and relate these functions to the model.
Building Functions (F – BF):
18) Build new functions from existing functions.
 F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x
+ k) for specific values of k (both positive and negative); find the value of k given the
graphs. Experiment with cases and illustrate an explanation of the effects on the graph
using technology. Include recognizing even and odd functions from their graphs and
algebraic expressions for them.
Linear and Exponential Models (F – LE):
19) Construct and compare linear and exponential models and solve problems.
Page 23 of 144

F- LE 3: Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a
polynomial function.
Quantities (N-Q):
20) Reason quantitatively and use units to solve problems.
 N – Q 2: Define appropriate quantities for the purpose of descriptive modeling.
Interpreting Categorical and Quantitative Data (S – ID):
21) Summarize, represent, and interpret data on two categorical and quantitative
variables.
 S – ID 6a: Fit a function to the data; use functions fitted to data to solve problems in the
context of the data. Use given functions or choose a function suggested by the context.
Emphasize linear and exponential models.

S – ID 6b: Informally assess the fit of a function by plotting and analyzing residuals.
Reasoning with Equations and Inequalities (A – REI):
22) Solve systems of equations.
 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.
Modeling Unit and Project –(10 days)
Page 24 of 144
High School Common Core Math II
3rd Nine Weeks
Scope and Sequence
Topic 5–Quadratic Functions – Solving by factoring (20 days)
Arithmetic with Polynomials and Rational Expressions (A – APR):
23) Perform arithmetic operations on polynomials.
 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.
Reasoning with Equations and Inequalities (A – REI):
24) Understand solving equations as a process of reasoning and explain the reasoning.
 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.
Reasoning with Equations and Inequalities (A – REI):
25) Solve equations and inequalities in one variable.
 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.
Interpreting Functions (F – IF):
26) Analyze functions using different representations.
 F – IF 8a: Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and interpret these in
terms of a context.
Creating Equations (A – CED):
27) Create equations that describe numbers or relationships.
 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.
Seeing Structure in Expressions (A – SSE):
28) Interpret the structure of expressions.
 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 factor not
depending on P.
Seeing Structure in Expressions (A – SSE):
29) Write expressions in equivalent forms to solve problems.
 A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines.
Topic 6– Quadratic Functions – Completing the Square/Quadratic Formula (20 days)
Page 25 of 144
Reasoning with Equations and Inequalities (A – REI):
30) Understand solving equations as a process of reasoning and explain the reasoning.
 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.
Reasoning with Equations and Inequalities (A – REI):
31) Solve equations and inequalities in one variable.
 A – REI 4: Solve quadratic equations in one variable.

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.

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.
Seeing Structure in Expressions (A – SSE):
32) Write expressions in equivalent forms to solve problems.
 A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Interpreting Functions (F – IF):
33) Analyze functions using different representations.
 F – IF 8: Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.

F – IF 8a: Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and interpret these in
terms of a context.
Creating Equations (A – CED):
34) Create equations that describe numbers or relationships.
 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.
The Complex Number System (N – CN):
35) Perform arithmetic operations with complex numbers.
 N – CN 1: Know there is a complex number i such that i 2  1 , and every complex
number has the form a+bi with a and b real.

N – CN 2: Use the relation i 2  1 and the commutative, associative, and distributive
properties to add, subtract, and multiply complex numbers.
The Complex Number System (N – CN):
36) Use complex numbers in polynomial identities and equations.
Page 26 of 144

N – CN 7: Solve quadratic equations with real coefficients that have complex solutions.
Page 27 of 144
High School Common Core Math II
4th Nine Weeks
Scope and Sequence
Topic 7 –Probability (20 days)
Conditional Probability and the Rules of Probability (S – CP):
37) Understand independence and conditional probability and use them to interpret
data.
 S – CP 1: Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or,” “and,” “not”).

S – CP 2: Understand that two events A and B are independent if the probability of A and
B occurring together is the product of their probabilities, and use this characterization to
determine if they are independent.

S – CP 3: Understand the conditional probability of A given B as P(A and B)/P(B), and
interpret independence of A and B as saying that the conditional probability of A given B
is the same as the probability of A, and the conditional probability of B given A is the
same as the probability of B.

S – CP 4: Construct and interpret two-way frequency tables of data when two categories
are associated with each object being classified. Use the two-way table as a sample space
to decide if events are independent and to approximate conditional probabilities. For
example, collect data from a random sample of students in your school on their favorite
subject among math, science, and English. Estimate the probability that a randomly
selected student from you school will favor science given that the student is in the tenth
grade. Do the same for other subjects and compare the results.

S – CP 5: Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. For example, compare the
chance of having lung cancer if you are a smoker with the chance of being a smoker if
you have lung cancer.
Conditional Probability and the Rules of Probability (S – CP):
38) Use the rules of probability to compute probabilities of compound events in a
uniform probability model.
 S – CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes
that also belong to A, and interpret the answer in terms of the model.

S – CP 7: Apply the Addition Rule, P(A or B) = P(B) – P(A and B), and interpret the
answer in terms of the model.
Topic 8 – Geometric Measurement (10 days)
Geometric Measurement and Dimension (G – GMD):
39) Explain volume formulas and use them to solve problems.
 G – GMD 1: Give an informal argument for the formulas for the circumference of a
circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
Page 28 of 144

G – GMD 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
Geometric and Modeling Project-(15 days)
*Modeling with Geometry (G – MG):
40) Apply geometric concepts in modeling situations.
 G – MG 1*: Use geometric shapes, their measures, and their properties to describe
objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

G – MG 2*: Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).*

G – MG 3*: Apply geometric methods to solve design problems (e.g., designing an
object or structure to satisfy physical constraints or minimize cost; working with
typographic grid systems based on ratios).*
Page 29 of 144
COLUMBUS CITY SCHOOLS
HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE
Topic 5
CONCEPTUAL CATEGORY
TIME
Quadratic Functions: Solving by
Algebra and Functions
RANGE
20 days
Factoring A –CED 1, A – REI 4b, F
– IF 8a, A – SSE 1b, 3a, A – APR 1
Domain: Arithmetic with Polynomials and Rational Expressions (A – APR):
Cluster
23) Perform arithmetic operations on polynomials.
GRADING
PERIOD
3
Domain: Reasoning with Equations and Inequalities (A – REI):
Cluster
24) Understand solving equations as a process of reasoning and explain the reasoning.
25) Solve equations and inequalities in one variable.
Domain: Interpreting Functions (F – IF):
Cluster
26) Analyze functions using different representations.
Domain: Creating Equations (A – CED):
Cluster
27) Create equations that describe numbers or relationships.
Domain: Seeing Structure in Expressions (A – SSE):
Cluster
28) Interpret the structure of expressions.
29) Write expressions in equivalent forms to solve problems.
Standards
23) Perform arithmetic operations on polynomials.
 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.
24) Understand solving equations as a process of reasoning and explain the reasoning.
 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.
25) Solve equations and Inequalities in one variable.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 30 of 144
Columbus City Schools
12/1/13

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.
26) Analyze functions using different representations.
 F – IF 8a: Use the process of factoring and completing the square in a quadratic
function to show zero, extreme values, and symmetry of the graph, and interpret these in
terms of a context.
27) Create equations that describe numbers or relationships.
 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.
28) Interpret the structure of expressions.
 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 factor not
depending on P.
29) Write expressions in equivalent forms to solve problems.
 A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 31 of 144
Columbus City Schools
12/1/13
TEACHING TOOLS
Vocabulary: binomial, degree, factored form of a quadratic function, factoring, factors, FOIL
method, function, leading coefficient, like terms, monomial, parabola, polynomial, quadratic,
quadratic equation, quadratic function, quadratic inequality, quadratic term, roots, solutions,
Square Root Property, trinomial, Zero Product Property, zeros
Teacher Notes:
Factoring Polynomials
Factoring, a method of breaking down polynomials into their parts, can be used to solve
Quadratic equations. Follow the steps listed below to solve the quadratic ax2 + bx + c = 0; a  0.
Step 1 - Factor out any common factors.
Step 2 - Factor the remaining expression by determining which two integers when added = b,
and when multiplied = c.
Step 3 - Use the Principle of Zero Products, if ab = 0 then a = 0 or b = 0, to find the roots.
Example: Find the roots of 2x2+ 10x + 12 = 0 by factoring.
Step 1 - Factor out the common factor of 2.
2( x2 + 5 x + 6) = 0
Step 2 - Determine which integers when added = 5 and when multiplied = 6 by examining
the factor pairs of 6: 1, 6; -1, -6; 2, 3; -2 , -3. The sum of the factor pair 2, 3 is 5.
The quadratic equation factors as:
2( x + 2)( x + 3) = 0
Step 3 - Using the Principle of Zero Products we conclude that either 2 = 0, which is not
true, or x + 2 = 0 or x + 3 = 0. Solving these equations we find that the solutions
to the quadratic equation 2x2 + 10x + 12 = 0 are x = - 2 and x = -3.
When simplifying the quotient of two trinomials, factor two primes and cancel (numerator paired
to denominator) common factors.
x 2  2 x  3  x  3 x  1

x 2  7 x  12 ( x  3)( x  4) Factor the numerator and denominator.
x2  2 x  3 x  1

; x  3, 4 Since the denominator’s factors are (x – 3) and (x – 4), x  3 and
2
x  7 x  12 x  4
x  4. Otherwise the denominator would equal zero, making the
fraction undefined. The restrictions will need to be stated.
The Algebra 2 textbook covers solving quadratic equations by graphing, factoring, completing
the square, and the quadratic formula. You may wish to review multiplying binomials and
factoring quadratics. The factoring worksheets can be used as additional review; however, the
order is set up so that it lends itself to the “ac” method. The “ac” method utilizes factoring by
grouping.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 32 of 144
Columbus City Schools
12/1/13
Roots – Zeros
The roots or zeros of a polynomial function are those values of x that make the equation true
when set equal to zero, i.e. f(x) = 0 as shown below. Zero is the term used to describe the
solution of a polynomial function and root is the term used to describe the solution of a
polynomial equation.
f(x) = x2 + 5x + 6,
set f(x) = 0 giving the equation: x2 + 5x + 6 = 0,
factor: (x + 3)(x + 2) = 0
Solve: x = -3, x = -2
-3 and -2 are the roots of the equation.
Notice that the polynomial above has a degree of 2 and two roots.
Assume you want to factor 3x2 + 10x – 8.
1. Multiply the quadratic term and the constant term.
(3x2  -8 = -24x2)
2. Find the factors of product -24x2 that provide a sum of the linear term10x.
(12x + (-2x) = 10x)
3. Replace the linear term in the original expression with the factors of -24x2 that provide a
sum of 10x.
(3x2 + 12x – 2x – 8)
4. Factor the expression by grouping:
3x( x  4)  2( x  4)
( x  4)(3x  2)
This process will work with any quadratic expression!
Factoring polynomials
Always attempt to factor out what is common first. Here are some general guidelines of
factoring based on the number of terms.
Number of Terms
Any number
Technique
Greatest Monomial Factor
Two terms
Difference of squares
Difference of cubes
Sum of cubes
Three terms
Perfect square trinomial
Factoring a trinomial
Four terms or more
Factoring by grouping
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 33 of 144
Columbus City Schools
12/1/13
Suggestion: make a chart of this for your wall!
x2 – 49
x2 –7x + 7x – 49
x(x – 7) + 7(x – 7)
(x – 7)(x + 7)
Factors of - 49x2
(- 7x)(7x) = - 49x2
Sum of 0x
- 7x + 7x = 0x
4x2 + 16x + 15
4x2 + 6x + 10x + 15
2x(2x + 3) + 5(2x + 3)
(2x + 3)(2x + 5)
Factors of 60x2
(6x)(10x) = 60x2
Sum of 16x
6x + 10x = 16x
2x2 – 7x – 3
none of the
combinations
work
not factorable
Factors of – 6x2
(1x)(- 6x) = - 6x2
(- 1x)(6x) = - 6x2
(2x)(- 3x) = - 6x2
(- 2x)(3x) = - 6x2
Sum of – 7x
1x + (- 6x) = - 5x
- 1x + 6x = 5x
2x + (- 3x) = - 1x
- 2x + 3x = 1x
Teacher Notes for A-CED 1
http://www.purplemath.com/modules/ineqquad.htm
Written notes on solving quadratic inequalities can be found on this website.
Misconceptions/Challenges:
 Students make mistakes when factoring quadratic expressions, because they fail to
recognize the difference between when “a” is equal to one and when “a” is not equal to
one.
 Students make mistakes with arithmetic when factoring.
Instructional Strategies:
A – CED 1:
1) Provide students with a copy of “More Area Applications” (included in this Curriculum
Guide). Students will solve each problem by drawing a picture, writing an equation, and
finding the solution both algebraically and graphically.
A – REI 4b
1) Have students practice factoring to solve equations using the “Equation Cards” (included in
this Curriculum Guide).
Solving Factorable Quadratic Equations,
http://www.regentsprep.org/Regents/math/ALGEBRA/AE5/indexAE5.htm
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 34 of 144
Columbus City Schools
12/1/13
2) This website provides instruction for solving quadratic equations by factoring.
3) Practice: http://www.regentsprep.org/Regents/math/ALGEBRA/AE5/PFacEq.htm This
additional website has practice problems.
4) Quadratics: https://www.khanacademy.org/math/algebra/quadratics A series of links on
solving quadratics through factoring, completing the square, graphing, and the quadratic
equation are provided.
5) Solving a Quadratic Equation by Factoring:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%20
1:%20Solving%20a%20quadratic%20equation%20by%20factoring A video tutorial is
provided demonstrating how to solve a quadratic equation by use of factoring.
6) Solving a Quadratic Equation by Factoring:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%20
2:%20Solving%20a%20quadratic%20equation%20by%20factoring A video tutorial is
provided demonstrating how to solve a trinomial in the form ax2 + bx + c by use of
factoring.
7) Solving a Quadratic Equation by Factoring:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%20
3:%20Solving%20a%20quadratic%20equation%20by%20factoring A video tutorial is
provided demonstrating how to problem solve using factoring.
8) Solving Quadratic Equations by Factoring:
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Solving%20Quadratic%20F
actoring.pdf Students practice solving quadratic equations, written in different forms, by
factoring.
9) Solving Quadratic Equations by Factoring:
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Quadratic%20Equations%2
0By%20Factoring.pdf Students solve quadratic equations using factoring with the practice
problems found at this site.
10) Solving Quadratic Equations by Factoring:
http://www.montereyinstitute.org/courses/Algebra1/U09L2T2_RESOURCE/index.html
A warm up, video presentation, practice and review are provided as lessons on solving
quadratic equations by factoring.
11) Solve an Equation using the Zero Product Property: http://www.ixl.com/math/algebra1/solve-an-equation-using-the-zero-product-property Students are provided problems to
determine the solution of a quadratic equation by using the Zero Product Property. A
tutorial is provided is the solutions offered are incorrect.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 35 of 144
Columbus City Schools
12/1/13
12) Solve a Quadratic Equation by Factoringhttp://www.ixl.com/math/algebra-1/solve-aquadratic-equation-by-factoring: Students are provided problems to determine the solution
of a quadratic equation by factoring and using the zero product property. A tutorial is
provided is the solutions offered are incorrect.
F –IF 8a
1) Students will complete the activity “Factor Me If You Can” (included in this Curriculum
Guide) to connect solving by factoring and graphing.
2) Have students use the “Connecting Zeros, Roots, x-intercepts, and Solutions” worksheet
(included int his Curriculum Guide), to see the relationship between solutions obtained by
factoring and the x-intercepts or zeros of the quadratic function. The students should be able
to solve any quadratic function with real solutions graphically.
Students are to determine the minimum point, maximum point, roots and number of
solutions of various functions with and without technology in “Families of Graphs # 2”
(included in this Curriculum Guide).
3) Have the students use the “Solving Quadratics Graphically” activity (included in this
Curriculum Guide) to reinforce the connection between zeros and solutions.
4) Have students use the “Solving by Factoring” worksheet (included in this Curriculum
Guide) to practice solving quadratics by factoring.
5) Solve a Quadratic by Factoring: http://www.ixl.com/math/algebra-1/solve-a-quadraticequation-by-factoring This site offers a set of interactive practice problems and an
explanation for an incorrect solution.
6) Factoring Trinomials Part 1:
http://education.ti.com/en/us/activity/detail?id=E581F8E30F8A4C689F2A226A183FDC75
Students use technology to factor trinomials of the form x2 + bx + c, where b and c are
positive integers and relate factoring a quadratic trinomial to an area model.
7) Factoring Trinomials Part 2:
http://education.ti.com/en/us/activity/detail?id=1BEE8F88204147B6B8CD213556E97915
Students use technology to explore trinomials of the form x2 + bx + c, where b is negative
and c is positive using an area model to factor trinomials in this form.
8) Exploring Polynomials: Factors, Roots, and Zeros:
http://education.ti.com/en/us/activity/detail?id=384FB053735B4C86BBF76AA6E018891C
Students use graphing technology to discover the zeros of the linear factors are the zeros of
the polynomial function; connect the algebraic representation to the geometric
representation; and see the effects of a double and/or triple root on the graph of a cubic
function of the leading coefficient on a cubic function.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 36 of 144
Columbus City Schools
12/1/13
9) Zeros of a Quadratic Function:
http://education.ti.com/en/us/activity/detail?id=E9C63B78A29F47DFAA53DE57B74E212C
Students merge graphical and algebraic representations of a quadratic function and its linear
factors.
10) Factoring Trinomials (a = 1) (Easy):
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%201.pdf
Students practice factoring trinomials in the form ax2 + bx + c and ax2 + bx – c.
11) Factoring Trinomials (a  1) (Hard):
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%202.pdf
Students practice factoring trinomials in the form ax2 + bx + c and ax2 + bx – c.
12) Factoring Special Cases:
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20Special%20C
ases.pdf Students factor perfect square and difference of squares trinomials.
13) Factor by Grouping:
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20By%20Group
ing.pdf Students factor trinomials by grouping.
14) Factoring Trinomials:
http://www.algebrahelp.com/lessons/factoring/trinomial/
This site has written explanations for factoring quadratics.
15) Factoring Quadratics: The Simple Case:
http://www.purplemath.com/modules/factquad.htm
Students factor quadratics that looks like ax2 + bx + c where a is 1.
16) Factoring Quadratics: The Hard Case: The Modified "a-b-c" Method, or "Box":
http://www.purplemath.com/modules/factquad2.htm
Students factor trinomials that looks like ax2 + bx + c where a is not 1.
17) Factoring Perfect Square Trinomials – Ex 1:
http://patrickjmt.com/factoring-perfect-square-trinomials-ex1/
This site offers a video tutorial of a perfect square trinomial.
18) Factoring Perfect Square Trinomials – Ex 2:
http://patrickjmt.com/factoring-perfect-square-trinomials-ex-2/
This site offers a second video tutorial of a perfect square trinomial.
19) Factoring Perfect Square Trinomials – Ex 3:
http://patrickjmt.com/factoring-perfect-square-trinomials-ex3/
This site offers another video tutorial of a perfect square trinomial.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 37 of 144
Columbus City Schools
12/1/13
20) Factoring Trinomials: Factor by Grouping – Ex 1:
http://patrickjmt.com/factoring-trinomials-factor-by-grouping-ex-1/
This site offers a video tutorial for factoring a tutorial by grouping.
21) Factoring Trinomials: Factor by Grouping – Ex 2:
http://patrickjmt.com/factoring-trinomials-factor-by-grouping-ex-2/
This site offers a second video tutorial for factoring a tutorial by grouping.
22) Factoring Trinomials: Factor by Grouping – Ex 3:
http://patrickjmt.com/factoring-trinomials-factor-by-grouping-ex-3/
This site offers another video tutorial for factoring a tutorial by grouping.
23) Factoring Trinomials (A quadratic Trinomial) by Trial and Error:
http://patrickjmt.com/factoring-trinomials-a-quadratic-trinomial-by-trial-and-error/
This site offers a video tutorial for factoring using the technique of trial and error.
24) Factoring Trinomials by Trial and Error – Ex 2:
http://patrickjmt.com/factoring-trinomials-by-trial-and-error-ex-2/
This site offers another video tutorial for factoring using the technique of trial and error.
25) Solving Quadratic Equations by Factoring – Basic Examples:
http://patrickjmt.com/solving-quadratic-equations-by-factoring-basic-examples/
This site offers a video tutorial for solving quadratic equations by factoring.
26) Solving Quadratic Equations by Factoring – Another Example:
http://patrickjmt.com/solving-quadratic-equations-by-factoring-another-example/
This site offers another video tutorial for solving quadratic equations by factoring.
27) Factoring the Difference of Two Squares – Ex 1:
http://patrickjmt.com/factoring-the-difference-of-two-squares-ex-1/
This site offers a video tutorial for factoring the difference of two squares.
28) Factoring the Difference of Two Squares – Ex 2:
http://patrickjmt.com/factoring-the-difference-of-two-squares-ex-2-2/
This site offers a second video tutorial for factoring the difference of two squares.
29) Factoring the Difference of Two Squares – Ex 3:
http://patrickjmt.com/factoring-the-difference-of-two-squares-ex-3-2/
This site offers a third video tutorial for factoring the difference of two squares.
A –SSE 1b
1) Exploring Polynomials: Factors, Roots, and Zeros:
http://education.ti.com/en/us/activity/detail?id=384FB053735B4C86BBF76AA6E018891
C Students will investigate graphical and algebraic representations of a polynomial
function and its linear factors. They will determine the zeros of the polynomial function.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 38 of 144
Columbus City Schools
12/1/13
2) Interpreting Algebraic Expressions:
http://map.mathshell.org/materials/download.php?fileid=694
In this lesson students will translate between words, symbols, tables and area
representations of algebraic expressions.
3) I Rule:
http://www.uen.org/core/math/downloads/sec2_i_rule.pdf
In this lesson students examine quadratic patterns in multiple representations.
4) I Rule:
http://www.uen.org/core/math/downloads/sec2_i_rule_tn.pdf
This site offers teacher notes for the lesson.
5) Look Out Below:
http://www.uen.org/core/math/downloads/sec2_look_out_below.pdf
In this lesson students examine quadratic functions on various sized intervals to determine
average rates of change.
6) Look Out Below:
http://www.uen.org/core/math/downloads/sec2_look_out_below_tn.pdf
This site offers teacher notes for the lesson.
7) Something to Talk About:
http://www.uen.org/core/math/downloads/sec2_something_to_talk_about.pdf
In this lesson student are introduced to quadratic functions, designed to elicit
representations and surface a new type of pattern and change.
8) Something to Talk About:
http://www.uen.org/core/math/downloads/sec2_something_to_talk_about_tn.pdf
Teacher notes are provided for this lesson.
A – SSE 3a
1) Factoring Fanatic:
http://alex.state.al.us/lesson_view.php?id=4152
In this lesson, students are provided practice for finding the correct factors for trinomial.
They are provided with a Tic-Tac sheet to help them determine the pattern between the two
numbers.
2) Math.A-SSE.3a:
http://www.shmoop.com/common-core-standards/ccss-hs-a-sse-3a.html
Written instructions for solving quadratic equations by factoring can be found at this site.
3) Learning Progression for CCSSM A-SSE 3a:
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=8&ved=0CF
QQFjAH&url=http%3A%2F%2Foursland.edublogs.org%2Ffiles%2F2013%2F06%2FLearn
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 39 of 144
Columbus City Schools
12/1/13
ing-Progression-for-CCSSM-A-SSE.3a-HW11cs90f0.doc&ei=BnQCUrG9JpKCyAHe2YDYBw&usg=AFQjCNFpNq4CRjVE8YttcE0zih
jOJfWwEQ&sig2=X168Hq2ME_eUazensjhcPw&bvm=bv.50310824,d.aWc
This document provides instruction on solving quadratic equations by factoring and a
problem concerning suspension bridges.
4) Challenging Factoring Quadratics: https://app.activateinstruction.org/playlist/resourcesview/id/5036aaa7efea65014c000022/rid/5021ad53efea65235f000a27/bc0/explore/bc1/playl
ist
A student practice sheet for solving by quadratics is provided at this site.
A – APR 1
1) Polynomial Puzzler:
http://illuminations.nctm.org/LessonDetail.aspx?id=L798
In this activity Students solve polynomials by solving a puzzle. Students will factor
polynomials and multiply monomials and binomials.
2) Factoring Trinomials when (a = 1)
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%201.pdf
At this website students practice factoring trinomials.
3) Factoring Trinomials (a ≠ 1)
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%202.pdf
At this website students practice factoring trinomials.
4) Factoring Special Cases
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20Special%20C
ases.pdf
At this website students practice factoring special cases.
5) Factoring by Grouping
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20By%20Group
ing.pdf
At this website students practice factoring by grouping.
6) Factoring Quadratic Functions:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/factoringquadratic-expressions
A video tutorial on factoring quadratic expressions can be found at the site below.
7) Factoring Simple Quadratics:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/factoringpolynomials-1
A video tutorial provides an example of factoring simple quadratic equations.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 40 of 144
Columbus City Schools
12/1/13
8) Factoring Quadratic Expressions:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/factoringtrinomials-with-a-leading-1-coefficient
A video tutorial provides an explanation on factoring a trinomial expression.
9) Factoring Polynomials 1:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/e/factoring_pol
ynomials_1
This site provides interactive practice on factoring trinomials. If students need help, a
tutorial is provided.
10) A Geometric Investigation of (a + b)2
http://illuminations.nctm.org/Activity.aspx?id=4089
This geometric demonstration show the value of the square of the binomial (a + b).
11) An easy way to find the common monomial factor of a polynomial is to write the prime
factorization of each monomial and then identify the factors that are common to every
monomial and factor it out. Arrange students into groups of three and give each group a
polynomial that has three monomials. Each student takes one monomial and writes the
prime factorization for it. The group then compares the monomials and selects any prime
factors that are common to all three. The common factor will be the product of the selected
factors. Give each group one card. When the group has finished working with a polynomial
they can trade cards with another group. Have the groups continue trading cards until all
groups have found the common factor for all of the polynomials. Use the “Polynomial
Cards for Use with Prime Factorization” (included in this Curriculum Guide).
12) Use Algebra Tiles to model how to “Find the Greatest Common Monomial Factor Using
Algebra Tiles” (included in this Curriculum Guide). Work through several problems on the
overhead, while students work the same problems at their desks using the tiles (a recording
sheet with problems is in this Curriculum Guide). Students should write, in algebraic form,
what they are doing with the manipulatives to encourage making the connection between the
concrete and abstract models. Have students count out Algebra Tiles to represent the
polynomial to be factored. Students then arrange the tiles into a rectangle. Students should
be led, if necessary, to arrange the tiles into the most compact rectangle possible, this will
ensure that one of the factors is the greatest common factor. See the following example:


To factor 2x2 + 6x, first count out 2 x2-tiles and 6 x-tiles. Then arrange them into as
compact a rectangle as possible.
Then, look at the width and length of the rectangle to find the factors. The width is 2x
and the length is x + 3, therefore 2x2 + 6x can be rewritten as 2x(x + 3). This polynomial
could also have been arranged into a rectangle with length and width of x and 2x + 6,
however that would not have given the greatest common factor of 2x as one of the
dimensions.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 41 of 144
Columbus City Schools
12/1/13
13) Another method for factoring polynomials is factoring by grouping. Sometimes,
polynomials can be factored by grouping terms. A polynomial may have a common factor
that is a binomial. For example, 6x2 + 3x – 4x – 2 can be rewritten as 3x (2x + 1) – 2(2x + 1)
where 2x + 1 is a common factor that is a binomial. We can use the Distributive Property to
write 3x (2x + 1) – 2(2x + 1) as (2x + 1) (3x – 2). You can visualize this factoring procedure
with a geometric model. The model shows the same total area using the two different
arrangements. To make a model of 6x2 + 3x – 4x – 2, select Algebra Tiles and arrange them
into a rectangle. Then look at the width and length of the rectangle to find the factors.
These are the same factors found when factoring by grouping.
14) Use “Polynomial Cards for Factoring by Grouping” (included in this Curriculum Guide),
so that students can work with a partner to factor and model the problems. Students should
also find the simplified product for each polynomial (e.g., 6x2 + 3x – 4x – 2 = 6x2 – x – 2).
This will enable students to take polynomials in a simplified form and rewrite them so that
they can be factored using the grouping method.
15) Use “Drawkcab Problems” (included in this Curriculum Guide): Another method for
factoring trinomials of the form ax2 + bx + c is to work the multiplication process backwards.
This method incorporates the factoring by grouping method. To help students begin
developing an understanding of the process involved for this method, give students a
trinomial that can be factored. When using Algebra Tiles, students are required to add zero
pairs to make the product rectangle.
16) Students will complete the activity “Discovering the Difference of Two Squares” (included
in this Curriculum Guide) to discover the pattern for factoring a difference of two squares.
17) Have the students use “Factoring Using the Greatest Common Factor” activity (included
in this Curriculum Guide).
18) Have the students use “Factoring by Grouping”, and the “Factoring Worksheet” activities
(included in this Curriculum Guide).
19) Have students practice working backwards using Algebra Tiles to make zero pairs to make
the product rectangle.
Factor the following trinomial: x2 – 2x – 15.
 Step One – Place tiles that represent the trinomial on an Algebra Tile Mat. Place the
unit tiles so that they form a rectangle. This will allow you to finish the larger
rectangle using zero pairs of tiles
 Step Two – add zero pairs of Algebra Tiles (those tiles are outlined) so that they
complete the rectangle. The sides of the rectangle are the factors for the trinomial.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 42 of 144
Columbus City Schools
12/1/13
Step One
Step Two
 The factors would be x + 3 and x – 5.
Once students feel comfortable adding in the zero pairs, use the same trinomials to develop the
steps for factoring by working backwards. Show students an example (see below) of
multiplying a pair of factors that were found using the Algebra Tiles.
(x + 3)(x – 5)
x – 5x + 3x – 15
x2 – 2x – 15
2
Encourage students to notice the relationship between the coefficients of the x terms in the
second two steps. Once students have realized that the coefficient of the x term in the product is
found by adding or subtracting numbers that are a factor pair for the product of the coefficient of
x2 and the constant term, they can be taught the following general steps for factoring a trinomial
of the form ax2 + bx + c.
To factor 2x2 – 5x – 12 (a trinomial of the form ax2 + bx + c) follow these steps:
a. Find the product of (ax2) and (c)
(ax2)(c) = (2x2) (-12) = -24x2
b. Find a pair of factors of (a) (c) (x2) that have the sum of bx
Factor pairs
Sum
(-8x)(3x) = -24x2
-8x + 3x = -5x
c. Rewrite the polynomial, expressing bx as the sum of a factor pair.
2x2 – 8x + 3x – 12
30)
d. Use factoring by grouping to remove the GCF from the first two terms, and the GCF
from the last two terms. Then use the distributive property to write as a product of two
binomial factors.
2x2 – 8x + 3x – 12 = 2x(x – 4) + 3(x – 4) = (2x + 3) (x – 4)
Reteach/Extension
Reteach:
1) Solving Quadratic Equations:
http://advancedCCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 43 of 144
Columbus City Schools
12/1/13
algebra.gcs.hs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4
45175&fid=1732539&sessionid=
This is a re-teach practice sheet with an answer key provided on solving quadratic equations
by graphing and factoring.
2) Solving Quadratic Equations:
http://advancedalgebra.gcs.hs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=445175&fid=1
732539&sessionid=
This is a re-teach practice sheet with an answer key provided on solving quadratic equations
by graphing and factoring.
Extensions:
1) Performance Task:
http://insidemathematics.org/common-core-math-tasks/high-school/HS-A2006%20Quadratic2006.pdf
Students will find graphical properties of a quadratic function given by its formula and will
have to factor for some problems.
2) Performance Task:
http://insidemathematics.org/common-core-math-tasks/high-school/HS-F2007%20Graphs2007.pdf
This problem involves working with linear and quadratic functions and their
graphs and equations. Students will solve by factoring to justify their answer.
Textbook References
Textbook:
Algebra I, Glencoe (2005): pp. 481-486, 487-488, 489-494, 495-500, 501506,
509-514, 840, 841
Supplemental: Algebra I, Glencoe (2005):
Chapter 9 Resource Masters
Reading to Learn Mathematics, pp. 533, 539, 545, 551, 557
Study Guide and Intervention, pp. 529-530, 535-536, 541-542, 547-548, 553-554
Skills Practice, pp. 531, 537, 543, 549, 555
Practice, pp. 532, 538, 544, 550, 556
Enrichment, pp. 534, 546, 558
Textbook:
Algebra 2,Glencoe (2003): pp. 239-244, 270-275, 301-305, 837
Supplemental: Algebra 2,Glencoe (2003):
Chapter 5 Resource Masters
Reading to Learn Mathematics, pp. 261, 291
Study Guide and Intervention, pp. 257-258, 28-288
Skills Practice, pp. 259, 289
Practice, pp. 260, 290
Enrichment, pp. 262, 292
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 44 of 144
Columbus City Schools
12/1/13
Chapter 6 Resource Masters
Reading to Learn Mathematics, pp. 329
Study Guide and Intervention, pp. 325-326
Skills Practice, pp. 327
Practice, pp. 328
Textbook: Integrated Mathematics: Course 3, McDougal Littell (2002): pp. 45-52, 72-73, 645
Textbook: Advanced Mathematical Concepts, Glencoe (2004): pp. 141, 159-16, 169-170
Textbook: Mathematics II Common Core, Pearson, pp. 665-671, 672 – 678, 679 – 687, 688 –
694, 695 – 704, 738-740.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 45 of 144
Columbus City Schools
12/1/13
A – CED 1
Name_______________________________________________ Date___________ Period_____
More Area Application
For each problem below, draw a picture, write an equation, solve the problem algebraically, and
support your work graphically.
1. Given a rectangle with an area of 45 cm2, find the dimensions of the rectangle if the length is 4
cm more than the width.
2. Given a triangle with an area of 16 in2, find the height of the triangle if it is twice the length of
the base.
3. Given a circle with an area of 30 m2, find the radius and circumference of the circle.
4. Rectangle #1 has a length that is 5 less than twice a number and a width of 4 more than that
number. Rectangle #2 has a length of 1 less than the number and the width is the number. Find
the value of the number if the area of Rectangle #1 is equal to the area of Rectangle #2.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 46 of 144
Columbus City Schools
12/1/13
A – CED 1
Name_______________________________________________ Date___________ Period_____
More Area Application
Answer Key
1. Given a rectangle with an area of 45 cm2, find the dimensions of the rectangle if the length is 4
cm more than the width.
w
w+4
45 = w(w + 4)
45=w2 + 4w
w2 + 4w – 45 = 0
(w + 9)(w – 5) = 0
w = (-9, 5)
The width cannot be negative so w = 5.
The length is w + 4 = 5 + 4 = 9
2. Given a right triangle with an area of 16 in2, find the height of the triangle if it is twice the
length of the base.
16 = 12 b • 2b
16 = b2
4 = b
The base cannot be negative, so the height is 8.
2b
b
3. Given a circle with an area of 30 m2, find the radius and circumference of the circle.
30 = r2
C = 2r
2
30/ = r
C = 2(3.09)
r = 3.09
C = 19.4
4. Rectangle #1 has a length that is 5 less than twice a number and a width of 4 more than that
number. Rectangle #2 has a length of 1 less than the number and the width is the number. Find
the value of the number if the area of Rectangle #1 is equal to the area of Rectangle #2.
Rectangle #2
Rectangle #1
x+4
x–1
2x – 5
(2x – 5)(x + 4) = x(x – 1)
x
2x2 + 3x – 20 = x2 – x
x2 + 4x – 20 = 0
x = 2.9, -6.9
The dimensions of a rectangle cannot be negative so the only reasonable answer is 2.9
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 47 of 144
Columbus City Schools
12/1/13
A – REI 4b
Name_______________________________________________ Date___________ Period_____
Equation Cards
(to be solved by factoring)
x2 – x – 20 = 0
x2 + 9x + 18 = 0
2x2 + 9x – 5 = 0
6x2 + 7x = 20
2x2 – 15x = 27
x2 = 7x – 12
12x2 – 2x – 4 = 0
x(4x + 1) = 5
x(15x + 1) – 2 = 0
x(125 – x) = 2500
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 48 of 144
Columbus City Schools
12/1/13
A – REI 4b
Name_______________________________________________ Date___________ Period_____
Equation Cards to be solved by factoring
Answer Key
x2 – x – 20 = 0
x2 + 9x + 18 = 0
(x – 5)(x + 4) = 0
x = 5 and x = -4
(x + 6)(x + 3) = 0
x = -6 and x = -3
2x2 + 9x – 5 = 0
6x2 + 7x = 20
(x + 5)(2x – 1) = 0
x = -5 and x = 1/2
(3x – 4)(2x + 5) = 0
x = 4/3 and x = -5/2
2x2 – 15x = 27
x2 = 7x – 12
(2x + 3)(x – 9) = 0
x = -3/2 and x = 9
(x – 4)(x – 3) = 0
x = 4 and x = 3
12x2 – 2x – 4 = 0
x(4x + 1) = 5
(4x + 2)(3x – 2) = 0
x = -1/2 and x = 2/3
(x – 1)(4x + 5) = 0
x = 1 and x = -5/4
x(15x + 1) – 2 = 0
x(125 – x) = 2500
(3x – 1)(5x + 2) = 0
x = 1/3 and x = -2/5
(x – 25)(x – 100) = 0
x = 25 and x = 100
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 49 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Factor Me If You Can
Find the zeros of the following quadratic functions by factoring.
1. y = x² - 8x + 7
2. y = x² + 2x – 8
3. y = x² + 6x + 9
4. y = x² + 6x + 8
5. y = x² - 2x + 1
6. y = x² + 5x + 4
Use the graphing calculator to verify your answer. Sketch each equation on the grids provided
below. Use trace to find the x-intercepts graphically.
Find the zeros of the following functions by factoring if possible.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 50 of 144
Columbus City Schools
12/1/13
7. y = x² - 7x – 8
8. y = x² + 3x + 5
9. y = x² + 6x – 7
10. y = x² + 3x + 6
11. y = x² + 5
12. y = x² + 4x
Use the graphing calculator to verify your answer. Sketch the graphs of each of the functions.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 51 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Factor Me If You Can
Answer Key
Find the zeros of the following quadratic functions by factoring.
1. y = x² - 8x + 7
(x – 1)(x – 7)
2. y = x² + 2x – 8
(x + 4)(x – 2)
3. y = x² + 6x + 9
(x + 3)(x + 3)
4 y = x² + 6x + 8
(x + 4)(x + 2)
5. y = x² - 2x + 1
(x – 1)(x – 1)
6. y = x² + 5x + 4
(x + 4)(x + 1)
Use the graphing calculator to verify your answer. Sketch each equation on the grids provided
below. Use trace to find the x-intercepts graphically.
Find the zeros of the following functions by factoring if possible.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 52 of 144
Columbus City Schools
12/1/13
7. y = x² - 7x – 8
(x – 8)(x + 1)
8. y = x² + 3x + 5
Not factorable
9. y = x² + 6x – 7
(x + 7)(x – 1)
10. y = x² + 3x + 6
Not factorable
11. y = x² + 5
Not factorable
12. y = x² + 4x
x (x + 4)
Use the graphing calculator to verify your answer. Sketch the graphs of each of the functions.
7.
8.
9.
10
24
21
18
15
12
9
6
3
-25 -20 -15 -10 -5
-3
-6
-9
-12
-15
-18
-21
-24
6
4
2
5 10 15 20 25
-10 -8 -6
-4 -2
2
4
6
8
10
-20 -16 -12 -8 -4
-3
-6
-9
-12
-15
-18
-2
-4
-6
-8
11.
8
12 16 20
2
4
6
12.
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
-4 -2
4
-10
10.
-10 -8 -6
18
15
12
9
6
3
8
2
4
6
8
10
-10 -8 -6
-4 -2
2
4
6
8
10
-10 -8 -6
-4 -2
-2
-2
-2
-4
-4
-4
-6
-6
-6
-8
-8
-8
-10
-10
-10
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 53 of 144
8
10
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Connecting Zeros, Roots, x-intercepts and Solutions
A) Graph the following quadratic equations, identify the x-intercepts.
B) Solve the quadratic equation by factoring.
1. A) y  x2  3x 10
2. A) y  - x2  7 x  6
x-intercept(s):
x-intercept(s):
B) x2  3x  10  0
B) - x2  7 x  6  0
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 54 of 144
Columbus City Schools
12/1/13
3. A) y  x2  4x  4
4. A) y  5x2 15x
x-intercept(s):
B) x2  4 x  4  0
x-intercept(s):
B) 5x2 15x  0
What do you notice about the x-intercepts and the solutions you obtained by factoring?
The terms zeros, roots, and solutions are used interchangeably when solving equations.
The
of the graph of a function are the
of the equation f  x   0 . These numbers are called the
function. Solutions are also called
of the
.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 55 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Connecting Zeros, Roots, x-intercepts and Solutions
Answer Key
A) Graph the following quadratic equations, identify the x-intercepts.
B) Solve the quadratic equation by factoring.
1. A) y  x2  3x 10
2. A) y  - x2  7 x  6
x
y
x
y
-2
-1
0
1
1.5
2
3
4
5
0
-6
- 10
- 12
- 12.25
- 12
- 10
-6
0
-7
-6
-5
-4
- 3.5
-3
-2
-1
0
-6
0
4
6
6.25
6
4
0
-6
x-intercept(s): (- 2, 0) and (5, 0)
x-intercept(s): (- 6, 0) and (- 1, 0)
B) x2  3x  10  0
(x – 5)(x + 2) = 0
x – 5 = 0 or x + 2 = 0
x = 5 or x = - 2
B) - x2  7 x  6  0
- (x2 + 7x + 6) = 0
- (x + 6)(x + 1) = 0
(x + 6) = 0 or x + 1 = 0
x = - 6 or x = - 1
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 56 of 144
Columbus City Schools
12/1/13
3. A) y  x2  4x  4
x
y
0
1
2
3
4
4
1
0
1
4
x-intercept(s):
B) x2  4 x  4  0
(x – 2)2 = 0
x–2=0
x=2
4. A) y  5x2 15x
(2,0)
x
y
0
1
1.5
2
3
0
- 10
- 11.25
- 10
0
x-intercept(s): (0,0) and (3,0)
B) 5x2 15x  0
5x(x – 3) = 0
5x = 0 or x – 3 = 0
x = 0 or x = 3
What do you notice about the x-intercepts and the solutions you obtained by factoring?
The x-coordinates of the x-intercepts are the same as the solutions obtained when solving for
x. When solving the equations, you are trying to determine which x-values will give you a yvalue of zero. All x-intercepts will have a y-coordinate of zero. Therefore, when solving an
equation, the solutions correspond to the x-intercepts.
The terms zeros, roots, and solutions are used interchangeably when solving equations.
The
x-intercepts
of the graph of a function are the
solutions
of the equation f  x   0 . These numbers are called the
zeros
of the
function. Solutions are also called
roots
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 57 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Families of Graphs #2
1. Using the Families of Graphs activity that you previously completed, fill in the table below.
Function
Minimum Point
Maximum Point
Roots
Number of
Solutions
2
y=x +2
y = x2 – 2
y = (x - 2)2
y = (x - 2)2 + 2
y = (x + 2)2
y = (x + 2)2 – 2
y = -x2
y = -x2 + 2
y = -(x – 2)2
y = -x2 – 4x + 4
Without graphing, determine the vertex, roots, and number of solutions for the following functions.
Show all work.
1. f(x) = x2 – 2x – 8
2. f(x) = 2x2 + 8x – 10
3. f(x) = -x2 + 6x – 6
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 58 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Families of Graphs #2
Answer Key
1. Using the Families of Graphs activity that you previously completed, fill in the table below.
Function
Minimum Point
Maximum Point
Roots
Number of
Solutions
2
y=x +2
(0, 2)
None
None
No real
Solutions
2
y=x -2
(0, -2)
None
2 real solutions
± 2
y = (x - 2)2
(2, 0)
None
2
1 real solution
y = (x - 2)2 + 2
(2, 2)
None
None
y = (x + 2)2
(-2, 0)
None
-2
No real
solutions
1 real solution
y = (x + 2)2 - 2
(-2, -2)
None
-2 ± 2
2 real solutions
y = -x2
None
(0, 0)
0
1 real solution
y = -x2 + 2
None
(0, 2)
± 2
2 real solutions
y = -(x – 2)2
None
(2, 0)
2
1 real solution
y = -x2 – 4x + 4
None
(2, -8)
-2 ± 2 2
2 real solutions
Without graphing, determine the vertex, roots, and number of solutions for the following functions.
Show all work.
1. f(x) = x2 – 2x – 8
Vertex: (1, -9); Roots: x = 4 and x = -2; 2 solutions
2. f(x) = 2x2 + 8x – 10
Vertex: (-2, -18); Roots: x = 1 and x = -5; 2 solutions
3. f(x) = -x2 + 6x – 6
Vertex (3, 3); Roots: x = 3  3 ; 2 solutions
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 59 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Solving Quadratics Graphically
Sketch a graph of each quadratic equation; state the vertex, domain and range, x-intercepts (if they
exist), and the y-intercept for each of the graphs. Solve each quadratic equation by factoring or the
square root method.
1. y  x2  2x  8
Vertex:
Range:
y-intercept:
2. y  -2x2  4x  2
Vertex:
Range:
y-intercept:
Solve: x2  2 x  8  0
Domain:
x-intercept(s):
Solve: -2 x2  4 x  2  0
Domain:
x-intercept(s):
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 60 of 144
Columbus City Schools
12/1/13
3. y  6x2  5x  4
Vertex:
Range:
y-intercept:
4. y  - x2  4
Vertex:
Range:
y-intercept
Solve: 6x2  5x  4  0
Domain:
x-intercept(s):
Solve: - x2  4  0
Domain:
x-intercept(s):
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 61 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Solving Quadratics Graphically
Answer Key
Sketch a graph of each quadratic equation; state the vertex, domain and range, x-intercepts (if they
exist), and the y-intercept for each of the graphs. Solve each quadratic equation by factoring or the
square root method.
1. y  x2  2x  8
x
y
-4
-3
-2
-1
0
1
2
0
-5
-8
-9
-8
-5
0
Vertex: (- 1, - 9)
Range:  - 9,  
Solve: x2  2 x  8  0
(x + 4)(x – 2) = 0
x + 4 = 0 or x – 2 = 0
x = - 4 or x = 2
Domain:
 - ,  
x-intercept(s): (- 4,0) and (2,0)
y-intercept: (0,- 8)
2. y  -2x2  4x  2
x
y
-3
-2
-1
0
1
-8
-2
0
-2
-8
Solve: -2 x2  4 x  2  0
- 2(x2 + 2x + 1) = 0
- 2(x + 1) 2 = 0
(x + 1) 2 = 0
x+1=0
x=-1
 - ,  
Vertex: (- 1,0)
Domain:
Range:  - , 0
y-intercept: (0,- 2)
x-intercept(s): (- 1,0)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 62 of 144
Columbus City Schools
12/1/13
y  6 x 2  5x  4
3.
x
y
-1
0
5
12
1
2
7
-4
1
-5
24
-3
10
Solve: 6x2  5x  4  0
(3x - 4)(2x + 1) = 0
3x – 4 = 0 or 2x + 1 = 0
4
1
x  or x  3
2
Domain:
 - ,  
Range:  - 5.0417,  
4 
 1 
x-intercept(s):  , 0  and  - , 0 
3 
 2 
y-intercept: (0,- 4)
1 
 5
Vertex:  , - 5  or  0.4167, - 5.0417 
24 
 12
4. y  - x2  4
x
y
-2
-1
0
1
2
-8
-5
-4
-5
-8
Solve: - x2  4  0
- x2 = 4
x2 = - 4
x   -4
no real solution
 - ,  
Vertex: (0,- 4)
Domain:
Range: x   - 4
x-intercept(s): no x-intercepts
y-intercept: (0,- 4)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 63 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Solving By Factoring
Solve each of the following equations by factoring.
1. x2 13x  36  0
2. x2  2x  63  0
3. x2  2 x  8  0
4. x2  5x  24
5. x2  9  10 x
6. x 2  16  0
7. 4x2  25  0
8. 4x2  x
9. x3  4 x  0
10. 4x2  7  29 x
11. x3 12x2  32x  0
12. 12 x2  10  7 x
2
13. 4x  4 x  3  0
14. x  6 x  1  12
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 64 of 144
Columbus City Schools
12/1/13
F – IF 8a
Name_______________________________________________ Date___________ Period_____
Solving by Factoring
Answer Key
Solve each of the following equations by factoring.
1. x2 13x  36  0
2. x2  2x  63  0
(x – 9)(x – 4) = 0
(x + 9)(x – 7) = 0
x = 9 or x = 4
x = - 9 or x = 7
3. x2  2 x  8  0
(x + 4)(x – 2) = 0
x = - 4 or x = 2
4. x2  5x  24
x2 – 5x – 24 = 0
x = 8 or x = - 3
5. x2  9  10 x
x2 – 10x + 9 = 0
(x – 9)(x – 1) = 0
x = 9 or x = 1
6. x 2  16  0
(x + 4)(x – 4) = 0
x = - 4 or x = 4
7. 4x2  25  0
(2x + 5)(2x – 5) = 0
8. 4x2  x
4x2 – x = 0
x(4x – 1) = 0
5
5
or x 
2
2
3
9. x  4 x  0
x(x2 – 4) = 0
x(x + 2)(x – 2) = 0
x = 0 or x + 2 = 0 or x – 2 = 0
1
4
2
10. 4x  7  29 x
4x2 – 29x + 7=0
(4x – 1)(x – 7) = 0
4x – 1 = 0 or x – 7 = 0
1
x  or x = 7
4
12. 12 x2  10  7 x
12x2 – 7x – 10 = 0
(3x + 2)(4x – 5) = 0
3x + 2 = 0 or 4x – 5 = 0
2
5
x   or x 
3
4
x
6
x

1

12

14. 
x
x = 0 or x = - 2 or x = 2
11. x3 12x2  32x  0
x(x2 – 12x + 32) = 0
x(x – 8)(x – 4) = 0
x = 0 or x – 8 = 0 or x – 4 = 0
x = 0 or x = 8 or x = 4
2
13. 4x  4 x  3  0
(2x – 1)(2x + 3) = 0
2x – 1 = 0 or 2x + 3 = 0
1
3
or x  
x
2
2
x = 0 or x 
6x2 – x – 12 = 0
(3x + 4)(2x – 3) = 0
3x + 4 = 0 or 2x – 3 = 0
x
4
3
or x 
3
2
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 65 of 144
Columbus City Schools
12/1/13
A– APR 1
Name_______________________________________________ Date___________ Period_____
Polynomial Cards
Use for Prime Factorization
Teacher note: Label the back of each card with a number or letter to make switching cards between
groups easier.
16x2y + 42xy2 – 20x2y2
24x3 + 32x2 – 48x
6y4 – 15y2 + 24y
4x2y + 12x2y2 + 20xy3
6x3 – 14x2 – 20x
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 66 of 144
Columbus City Schools
12/1/13
-3x5 + 15x3 + 6x2
8x – 6x – 18x
3
2
3x4 + 12x2 – 9x
3
2
12x + 8x + 20x
30y – 18y + 54y
5
3
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 67 of 144
2
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Polynomial Cards
Use for Prime Factorization
Answer Key
Polynomial: 16x2y + 42xy2 – 20x2y2
Prime Factorization: 16x2y factors to: 2 • 2 • 2 • 2 • x • x • y
42xy2 factors to: 2 • 3 • 7 • x • y • y
-20x2y2 factors to: -1 • 2 • 2 • 5 • x • x • y • y
GCF: 2xy
Polynomial written as two factors: (2xy)(8x + 21y – 10xy)
Polynomial: 24x3 + 32x2 – 48x
Prime Factorization: 24x3 factors to: 2 • 2 • 2 • 3 • x • x • x
32x2 factors to: 2 • 2 • 2 • 2 • 2 • x • x
-48x factors to: -1 • 2 • 2 • 2 • 2 • 3 • x
GCF: 8x
Polynomial written as two factors: (8x)(3x2 + 4x – 6)
Polynomial: 6y4 – 15y2 + 24y
Prime Factorization: 6y4 factors to: 2 • 3 • y • y • y • y
-15y2 factors to: -1 • 3 • 5 • y • y
24y factors to: 2 • 2 • 2 • 3 • y
GCF: 3y
Polynomial written as two factors: (3y)(2y3 – 5y + 8)
Polynomial: 4x2y + 12x2y2 + 20xy3
Prime Factorization: 4x2y factors to: 2 • 2 • x • x • y
12x2y2 factors to: 2 • 2 • 3 • x • x • y • y
20xy3 factors to: 2 • 2 • 5 • x • y • y • y
GCF: 4xy
Polynomial written as two factors: (4xy)(x + 3xy + 5y2)
Polynomial: 6x3 – 14x2 – 20x
Prime Factorization: 6x3 factors to: 2 • 3 • x • x • x
-14x2 factors to: -1 • 2 • 7 • x • x
-20x factors to: -1 • 2 • 2 • 5 • x
GCF: 2x
Polynomial written as two factors: (2x)(3x2 – 7x – 10)
Polynomial: -3x5 + 15x3 + 6x2
Prime Factorization: -3x5 factors to: -1 • 3 • x • x • x • x • x
15x3 factors to: 3 • 5 • x • x • x
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 68 of 144
Columbus City Schools
12/1/13
6x2 factors to: 2 • 3 • x • x
GCF: 3x2
Polynomial written as two factors: (3x2)(-x3 + 5x + 2)
Polynomial: -8x3 – 6x2 – 18x
Prime Factorization: -8x3 factors to: -1 • 2 • 2 • 2 • x • x • x
-6x2 factors to: -1 • 2 • 3 • x • x
-18x factors to: -1 • 2 • 3 • 3 • x
GCF: -2x
Polynomial written as two factors: (-2x)(4x2 + 3x + 9)
Polynomial: 3x4 + 12x2 – 9x
Prime Factorization: 3x4 factors to: 3 • x • x • x • x
12x2 factors to: 2 • 2 • 3 • x • x
-9x factors to: -1 • 3 • 3 • x
GCF: 3x
Polynomial written as two factors: (3x)(x3 + 4x – 3)
Polynomial: 12x3 + 8x2 + 20x
Prime Factorization: 12x3 factors to: 2 • 2 • 3 • x • x • x
8x2 factors to: 2 • 2 • 2 • x • x
20x factors to: 2 • 2 • 5 • x
GCF: 4x
Polynomial written as two factors: (4x)(3x2 + 2x + 5)
Polynomial: 30y5 – 18y3 + 54y2
Prime Factorization: 30y5 factors to: 2 • 3 • 5 • y • y • y • y • y
-18y3 factors to: -1 • 2 • 3 • 3 • y • y • y
54y2 factors to: 2 • 3 • 3 • 3 • y • y
2
GCF: 6y
Polynomial written as two factors: (6y2)(5y3 – 3y + 9)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 69 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Finding the Greatest Common Monomial Using
Algebra Tiles



First, count out Algebra Tiles to represent the polynomial.
Second, arrange the tiles into a rectangle. Sketch the rectangle on this sheet.
Third, look at the width and length of the rectangle. This represents the factors of the
polynomial.
1. 3x2 + 6x =
2. 2x2 – 3x =
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 70 of 144
Columbus City Schools
12/1/13
3. 3x2 – 15x =
4. 4x2 + 6x =
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 71 of 144
Columbus City Schools
12/1/13
5. 3x + 6 =
6. 2x2 – x =
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 72 of 144
Columbus City Schools
12/1/13
7. 4x2 + 12x =
8. –2x2 + 4x =
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 73 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Finding the Greatest Common Monomial
Using Algebra Tiles



Answer Key
First, count out Algebra Tiles to represent the polynomial.
Second, arrange the tiles into a rectangle. Sketch the rectangle on this sheet.
Third, look at the width and length of the rectangle. This represents the factors of the
polynomial.
1. 3x2 + 6x = 3x(x + 2)
2. 2x2 – 3x = x(2x – 3)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 74 of 144
Columbus City Schools
12/1/13
3. 3x2 – 15x = 3x(x – 5)
4. 4x2 + 6x = 2x(2x + 3)
5.
3x + 6 = 3(x + 2)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 75 of 144
Columbus City Schools
12/1/13
6. 2x2 – x = x(2x – 1)
7. 4x2 + 12x = 4x(x + 3)
8. –2x2 + 4x = -2x(x – 2)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 76 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Polynomial Cards
Use for Factoring by Grouping
Teacher note: Label the back of each card with a number or letter to make it easier for students to
record their work.
2
x + 2x + 7x + 14
x – 9x + 4x – 36
2
x2 + 2x + 3x + 6
x2 – 3x + 6x – 18
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 77 of 144
Columbus City Schools
12/1/13
6x2 – 3x + 4x – 2
3x – 6x – 4x + 8
2
2x2 – 6x + 4x – 12
x2 + 4x – 3x – 12
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 78 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Polynomial Cards
Use for Factoring by Grouping
Answer Key
x2 + 2x + 7x + 14
(x2 + 2x) + (7x + 14)
x(x + 2) + 7(x + 2)
(x + 7)(x + 2)
Simplified product: x2 + 9x + 14
Polynomial:
Factors:
x2 – 9x + 4x – 36
(x2 – 9x) + (4x – 36)
x(x – 9) + 4(x – 9)
(x + 4)(x – 9)
Simplified product: x2 – 5x – 36
Polynomial:
Factors:
x2 + 2x + 3x + 6
(x2 + 2x) + (3x + 6)
x(x + 2) + 3(x + 2)
(x + 3)(x + 2)
Simplified product: x2 + 5x + 6
Polynomial:
Factors:
x2 – 3x + 6x – 18
(x2 – 3x) + (6x – 18)
x(x – 3) + 6(x – 3)
(x + 6)(x – 3)
Simplified product: x2 + 3x – 18
Polynomial:
Factors:
6x2 – 3x + 4x – 2
(6x2 – 3x) + (4x – 2)
3x(2x – 1) + 2(2x – 1)
(3x + 2)(2x – 1)
Simplified product: 6x2 + 1x – 2
Polynomial:
Factors:
3x2 – 6x – 4x + 8
(3x2 – 6x) + (-4x + 8) or could be written as (3x2 – 6x) – (4x – 8)
3x(x – 2) – 4(x – 2)
(3x – 4)(x – 2)
Simplified product: 3x2 – 10x + 8
Polynomial:
2x2 – 6x + 4x – 12
Factors:
(2x2 – 6x) + (4x – 12)
2x(x – 3) + 4(x – 3)
(2x + 4)(x – 3)
Simplified product: 2x2 – 2x – 12
Polynomial:
Factors:
x2 + 4x – 3x – 12
x(x + 4) – 3(x + 4)
(x – 3)(x + 4)
Simplified product: x2 + x – 12
Polynomial:
Factors:
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 79 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Drawkcab Problems
(Backward Problems)
Factor each polynomial by working backwards. Use Algebra Tiles to make a model of the
polynomial.
1. x2 + 2x – 8 =
2. x2 + x – 6 =
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 80 of 144
Columbus City Schools
12/1/13
3. x2 – x – 12 =
4. x2 + 2x – 15 =
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 81 of 144
Columbus City Schools
12/1/13
5. x2 – 3x – 10 =
6. 2x2 – 9x + 4 =
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 82 of 144
Columbus City Schools
12/1/13
7. 6x2 + 17x + 5 =
8. 3x2 + 10x – 8 =
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 83 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Drawkcab Problems
(Backward Problems)
Answer Key
Factor each polynomial by working backwards. Use Algebra Tiles to make a model of the
polynomial.
1. x2 + 2x – 8 =(x + 4)(x – 2)
Teacher note: Zero pairs of Algebra
Tiles that were added to make a
complete rectangle are outlined to
make them more obvious. Students
may need to be guided to add these
zero pairs.
2. x2 + x – 6 = (x + 3)(x – 2)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 84 of 144
Columbus City Schools
12/1/13
3. x2 – x – 12 = (x + 3)(x – 4)
4. x2 + 2x – 15 = (x + 5)(x – 3)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 85 of 144
Columbus City Schools
12/1/13
5. x2 – 3x – 10 = (x – 5)(x + 2)
6. 2x2 – 9x + 4 = (2x – 1)(x – 4)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 86 of 144
Columbus City Schools
12/1/13
7. 6x2 + 17x + 5 = (3x + 1)(2x + 5)
8. 3x2 + 10x – 8 = (3x – 2)(x + 4)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 87 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Discovering the Difference of Two Squares
1.
Draw a square, using a ruler to measure each side, and label each side as “a”.
a
a
2.
Draw a smaller square inside the upper left corner of your current square, and label each
side as “b”.
b
a
b
a
3.
Shade the original square, leaving out the new square.
b
a
b
a
4.
Find an expression for the area of the shaded region, in terms of “a” and “b”.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 88 of 144
Columbus City Schools
12/1/13
5.
Cut the non-shaded area off, and determine an expression for each side of the remaining
figure.
b
b
a
a
a
a
6.
Cut the shaded area along the dotted line to make two separate rectangles, and then place
them together to form one rectangle, labeling each side of the new rectangle in terms of “a”
and “b”.
a
a
7.
Determine another expression for the area of the shaded region, using the new rectangle.

The shaded region has not been changed, just re-arranged, therefore the two expressions
must be equal.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 89 of 144
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A- – APR 1
Name_______________________________________________ Date___________ Period_____
Discovering the Difference of Two Squares
Answer Key
1.
Draw a square, using a ruler to measure each side, and label each side as “a”.
a
a
2.
Draw a smaller square inside the upper left corner of your current square, and label
each side as “b”.
b
a
b
a
3.
Shade the original square, leaving out the new square.
b
a
b
a
4.
Find an expression for the area of the shaded region, in terms of “a” and “b”.
A = a2 – b2
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 90 of 144
Columbus City Schools
12/1/13
5.
Cut the non-shaded area off, and determine an expression for each side of the remaining
figure, in terms of a and b.
a-b
b
b
b
a
b
a
a-b
a
6.
a
Cut the shaded area along the dotted line to make two separate rectangles, and then
place them together to form one rectangle, labeling each side of the new rectangle in
terms of “a” and “b”.
a-b
b
b
a-b
a
a-b
a
b
a
7.
Determine another expression for the area of the shaded region, using the new
rectangle.
A = (a – b)(a + b)

The shaded region has not been changed, just re-arranged, therefore the two expressions
must be equal.
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 91 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Factoring Using the Greatest Common Factor
Factor each polynomial as the product of its greatest common factor and another polynomial.
1. 6 x  12
2. 14 x 12
4. 4 x  8 y 12
5. 14s 2  21st
7. 8x3  16 x2
8. 15x2  9 x
3. 9 x2  6 x 12
6. 10x3  5x2  15x
9.  r 2h  2 r 2
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 92 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Factoring Using the Greatest Common Factor
Answer Key
Factor each polynomial as the product of its greatest common factor and another polynomial.
1. 6 x  12
2. 14 x 12
6(x + 2)
2(7x – 6)
4. 4 x  8 y 12
5. 14s 2  21st
4(x + 2y – 3)
7. 8x3  16 x2
8x2(x + 2)
7s(2s + 3t)
8. 15x2  9 x
3x(5x – 3)
3. 9 x2  6 x 12
3(3x2 + 2x – 4)
6. 10x3  5x2  15x
5x(2x2 – x + 3)
9.  r 2h  2 r 2
 r 2 (h  2)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 93 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Factoring By Grouping
Factor. Check by multiplying the factors.
1. 3  x  y   x  x  y 
2. 2 x  x  4   7  x  4 
3. 5 x  x  3   x  3
4.  5 x  4   x  5 x  4 
5. 6 x  2 x  1  5   2 x  1
6. 4 x  2  3 x   7  3 x  2 
7. pq  2qr  2r 2  pr
8. 6 x  3 y  2 xz  yz
9. ab  2b  ac  2c
10. x3  2 x2  3x  6
11. 2 x3  x2  8x  4
12. 2x3  6x2  5x  15
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 94 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Factoring By Grouping
Answer Key
Factor. Check by multiplying the factors.
1. 3  x  y   x  x  y 
(x + y)(3 + x)
2. 2 x  x  4   7  x  4 
(x – 4)(2x + 7)
3. 5 x  x  3   x  3
(x + 3)(5x – 1)
4.  5 x  4   x  5 x  4 
(5x + 4)(1 – x)
5. 6 x  2 x  1  5   2 x  1
6x(2x – 1) – 5(2x – 1)
(2x – 1)(6x – 5)
or
(- 2x + 1)(- 6x + 5)
6. 4 x  2  3 x   7  3 x  2 
4x(2 – 3x) + 7(- 3x + 2)
(2 – 3x)(4x + 7)
or
(3x – 2)(- 4x – 7)
7. pq  2qr  2r 2  pr
q(p + 2r) + r(2r + p)
(p + 2r)(q + r)
8. 6 x  3 y  2 xz  yz
3(2x – y) + z(2x – y)
(2x – y)(3 + z)
9. ab  2b  ac  2c
b(a – 2) + c(a – 2)
(a – 2)(b + c)
10. x3  2 x2  3x  6
x2(x – 2) + 3(x – 2)
(x – 2)(x2 + 3)
11. 2 x3  x2  8x  4
x2 (2x + 1) + 4(2x + 1)
(2x + 1)(x2 + 4)
12. 2x3  6x2  5x  15
2x2 (x – 3) – 5(x – 3)
(x – 3)(2x2 – 5)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 95 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Factoring Worksheet
Completely factor the following polynomials. Rewrite the problem, show all of your work and the
answer on a separate piece of paper.
1. 2 x2  5x  3
2. 7 x2  8x  1
3. x 2  4 x  5
4. x2  2 x  35
5. x2  12 x  24
6. 49 x2  81
7. 49  14x  x2
8. 81x4  16
9. 2x2  4x  2
10. 4 x 2  x  3
11. x2  3x  54
12. x2  15x  44
13. 64x2 16xy  y 2
14. 64 121c4
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 96 of 144
Columbus City Schools
12/1/13
15. x2 13x  42
16. x2  14 x  51
17. x2  20 x  51
18. x2  3x  40
19. 7 x2 18x  8
20. 10  x  2 x2
21. 6  23x  4 x2
22. 9x2  25x  6
23. 4 x2  12 x  9
24. x4  x2  56
25. 7 x2  19 x  6
26. 36 x2  5x  24
27. 144 x2  169
28. 20 x2  27 x  8
29. 12 x2  7 x 10
30. 3x2  7 x  6
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 97 of 144
Columbus City Schools
12/1/13
A – APR 1
Name_______________________________________________ Date___________ Period_____
Factoring Worksheet
Answer Key
Completely factor the following polynomials. Rewrite the problem; show all of your work and the
answer on a separate piece of paper.
1. 2 x2  5x  3
2. 7 x2  8x  1
(2x + 3)(x + 1)
(7x – 1)(x – 1)
3. x 2  4 x  5
(x + 5)(x – 1)
4. x2  2 x  35
(x + 7)(x – 5)
5. x2  12 x  24
prime
6. 49 x2  81
(7x + 9)(7x – 9)
7. 49  14x  x2
(7 – x)2
8. 81x4  16
(9x2 + 4)(9x2 – 4)
(9x2 + 4)(3x + 2)(3x – 2)
9. 2x2  4x  2
2(x2 – 2x + 1)
2(x – 1) 2
10. 4 x 2  x  3
(4x + 3)(x – 1)
11. x2  3x  54
(x + 9)(x – 6)
12. x2  15x  44
(x + 11)(x + 4)
13. 64x2 16xy  y 2
(8x – y) 2
14. 64 121c4
(8 + 11c2)(8 – 11c2)
15. x2 13x  42
(x – 6)(x – 7)
16. x2  14 x  51
(x + 17)(x – 3)
17. x2  20 x  51
(x + 17)(x + 3)
18. x2  3x  40
(x – 8)(x + 5)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 98 of 144
Columbus City Schools
12/1/13
19. 7 x2 18x  8
(7x – 4)(x – 2)
20. 10  x  2 x2
(5 – 2x)(2 + x)
21. 6  23x  4 x2
(6 + x)(1 – 4x)
22. 9x2  25x  6
(9x + 2)(x – 3)
23. 4 x2  12 x  9
(2x + 3) 2
24. x4  x2  56
(x2 – 8)(x2 + 7)
25. 7 x2  19 x  6
(7x – 2)(x + 3)
26. 36 x2  5x  24
(9x – 8)(4x + 3)
27. 144 x2  169
(12x + 13)(12x – 13)
28. 20 x2  27 x  8
(5x + 8)(4x – 1)
29. 12 x2  7 x 10
(4x – 5)(3x + 2)
30. 3x2  7 x  6
(3x + 2)(x – 3)
CCSSM II
Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b,
3a, A – APR 1
Quarter 3
Page 99 of 144
Columbus City Schools
12/1/13
COLUMBUS CITY SCHOOLS
HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE
Topic 6
CONCEPTUAL CATEGORY
TIME
GRADING
Quadratic Functions: Completing Algebra, Functions, Number
RANGE
PERIOD
20 days
the Square and the Quadratic
and Quantity
3
Formula: A – REI 1, 4, 4a, 4b; A –
SSE 3b; F – IF 8, 8a; A – CED 1;
N – CN 1, 2, 7
Domain: Reasoning with Equations and Inequalities (A – REI):
Cluster
30) Understand solving equations as a process of reasoning and explain the reasoning.
31) Solve equations and inequalities in one variable.
Domain: Seeing Structure in Expressions (A – SSE):
Cluster
32) Write expressions in equivalent forma to solve problems.
Domain: Interpreting Functions (F – IF):
Cluster
33) Analyze functions using different representations.
Domain: Creating Equations (A – CED):
Cluster
34) Create equations that describe numbers or relationships.
Domain: The Complex Number System (N – CN):
Cluster
35) Perform arithmetic operations with complex numbers.
36) Use complex numbers in polynomial identities and equations.
Standards
30) Understand solving equations as a process of reasoning and explain the reasoning.
 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.
31) Solve equations and Inequalities in one variable.
 A – REI 4: Solve quadratic equations in one variable.

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.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 100 of 144
Columbus City Schools
12/1/13

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.
32) Write expressions in equivalent forms to solve problems.
 A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
33) Analyze functions using different representations.
 F – IF 8: Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.

F – IF 8a: Use the process of factoring and completing the square in a quadratic function to
show zero, extreme values, and symmetry of the graph, and interpret these in terms of a
context.
34) Create equations that describe numbers or relationships
 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.
35) Perform arithmetic operations with complex numbers.
 N – CN 1: Know there is a complex number i such that i 2  1 , and every complex number
has the form a+bi with a and b real.

N – CN 2: Use the relation i 2  1 and the commutative, associative, and distributive
properties to add, subtract, and multiply complex numbers.
36) Use complex numbers in polynomial identities and equations.
 N – CN 7: Solve quadratic equations with real coefficients that have complex solutions.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 101 of 144
Columbus City Schools
12/1/13
TEACHING TOOLS
Vocabulary: binomial, coefficient, completing the square, complex conjugates, complex
number, complex roots, constant term, degree, discriminant, extraneous solution, function,
imaginary number, imaginary part, imaginary unit, leading coefficient, polynomial, principal
root, pure imaginary number, quadratic equation, quadratic, quadratic equation, Quadratic
Formula, quadratic function, quadratic inequality, quadratic term, real number, real part, roots,
solutions, Square Root Property, square roots, standard form, trinomial, zeros
Teacher Notes:
Note: In the previous topic students solved by factoring, and taking square roots. In this topic,
students will solve by graphing and the use of the Quadratic Formula.
The Algebra 2 textbook covers solving quadratic equations by graphing, factoring, completing
the square, and the quadratic formula. You may wish to review multiplying binomials and
factoring quadratics.
Quadratic Formula:
Another algebraic method of finding the roots of a quadratic equation is the quadratic formula:
−𝑏±√𝑏 2 −4𝑎𝑐
𝑥=
2𝑎
and 𝑎 ≠ 0.
, where a, b and c represent the same values as the a, b and c in y = ax2 + bx + c
Example:
Solve 2x2  6 x  3  0 by using the quadratic formula.
x
6  62  4  2  3 6  36  24 6  12


22
4
4
x
6  2 3 3  3
6  2 3 3  3

or x 

4
2
4
2
a = 2, b = 6, c = 3:
The nature of the solutions of a quadratic function can be determined by examining the value of
the discriminant; b2 – 4ac.
Discriminant
b2 – 4ac > 0
b2 – 4ac = 0
b2 – 4ac < 0
Number of Roots
2 real roots
1 real root with multiplicity
0 real roots. The roots are imaginary numbers.
The section on solving quadratics by using the quadratic formula includes both real and complex
solutions. You may wish to select problems with real solutions first. The complex number
system is discussed in chapter 5, section 9 of the Algebra 2 text. The mode on the TI-84 can be
changed to work with imaginary numbers. To change the mode from Real to Complex, press the
MODE key, move your cursor down to Real, press the right arrow key one time, then press
ENTER. Upon completion of the Complex Number section, you will need to resume work using
the quadratic formula.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 102 of 144
Columbus City Schools
12/1/13
Complex Numbers
Complex numbers take the standard form a + bi, where a and b are real numbers and i, the
imaginary part is equal to 1 . In a complex number, a is the real part and bi is the complex
part, thus real numbers are actually complex numbers with b = 0.
Powers of i
The values of the powers of i repeat in a regular pattern every 4th power.
i 1= i
2
i2 = i  i = 1  1  1  1
i3 = i2  i = 1 i =  i
2
i4 = i3  i =  i  i =  1  1   1  (1)  1
To find the value of a power of i greater than 4, divide the exponent by 4 and examine the
remainder. If the remainder is:
1234-
the power of i has the same value as i1 which is equal to i.
the power of i has the same value as i2 which is equal to  1.
the power of i has the same value as i3 which is equal to  i.
the power of i has the same value as i4 which is equal to 1.
Example:
Find the value of i22.
22  4  5 , remainder 2, therefore i22 = i2 = 1.
Example with the quadratic formula:
To find the nonreal zeros of the function f(x) = x2 + x + 1, set x2 + x + 1 = 0, and apply the
quadratic formula.
-1  1  4(1)(1) -1  -3

2
2
-1  i 3 -1
3

 i
2
2
2
x
Operations on Complex Numbers
When adding complex numbers, add the real parts and add the imaginary parts as shown in
example 1 and example 2 below.
Example 1:
(a + bi) + (c + di) = (a + c) + (b + d)i
Example 2:
(3 + 4i) + (6 + 2i) = 9 + 6i
To multiply complex numbers, use the same method as you would use when multiplying two
binomials.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 103 of 144
Columbus City Schools
12/1/13
Example 1: (a + bi)(c + di) = ac + adi + bci + bdi2 = ac + (ad + bc)i + -(bd), since i² = -1
Example 2: (2 + 3i)(4 + 5i) = 8 + 10i +12i + 15i2 = 8 + 22i + -15 = -7 + 22i
Complex Conjugates
The conjugate of a complex number is that number by which you can multiply a complex
number to obtain a real number. The conjugate of a + bi is a – bi. Just as in the case of a
difference of squares, multiplying a complex number by its conjugate causes the middle term
containing the complex part to drop out leaving the real parts.
(a + bi)(a – bi) = a2 – abi + abi – (bi)2 = a2 + b2
When the denominator of a fraction is a complex number, multiplying the numerator and
denominator by the complex conjugate, will rationalize the denominator as shown below.
3  2i (3  2i)(5  6i) 15  18i  10i  12i 2 27  8i



5  6i (5  6i)(5  6i)
25  36
61
Completing the Square
When a quadratic with real roots doesn’t appear to be factorable, it can be forced to factoring by
completing the square. Completing the square converts the left hand side of the equation into a
perfect square trinomial. After factoring, the solution to the quadratic can be found by taking the
square root of each side of the equation and solving for the variable as shown in the general case
and the example below.
To complete the square of the quadratic
subtract c from each side
2
x2 + bx + c = 0,
x2 + bx = -c,
2
2
b
add   to each side of the equation
2
b
b
x  bx     c    ,
2
2
factor the left hand side
b

b
 x    c    ,
2

 2
2
2
2
2
take the square root of each side of the equation
2
b

b
 x    c    ’
2

2
2
solving for x
b
b
x    c+  
2
2 ,
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 104 of 144
Columbus City Schools
12/1/13
2
b
b
x    c+  
2
2 .
Example:
Find the roots of the quadratic x2 + 4x – 9 = 0 by completing the square.
subtract - 9 from each side of the equation
x2 + 4x = 9,
2
4
add   to each side of the equation
2
factor the left hand side and simplify
the right hand side
take the square root of each side
solve for x
2
2
4
 4
x2  4 x     9    ,
2
 2
2
 x  2   13 ,
x  2   13 ,
x  2  13 ,
x  2  13 , and , x  2  13
Please note that if a is not equal to one, it must be factored out of the equation prior to
making the left hand side into a perfect square trinomial. This value must be taken into
consideration when adding it to both sides of the equation. (See the example below.)
Find the roots of the quadratic 2x2 + 4x – 9 = 0 by completing the square.
subtract - 9 from each side of the equation
2x2 + 4x = 9,
factor the 2 out of the left hand side of the equation 2(x2 + 2x) = 9
2
2
2
2
add   to each side of the equation
2
2
2
2(x + 2x) +   = 9 +   ,
2
2
factor the left hand side and simplify
the right hand side
divide both sides by the factor
take the square root of each side
2(x + 1)2 = 10,
solve for x
x = - 1 5,
x = - 1 + 5 , and x = - 1 -
2
(x + 1)2 = 5
x + 1 = 5,
5
Misconceptions/Challenges:
 Students make mistakes when evaluating the quadratic formula, because they do not
understand the difference between (b) 2 and  b 2 .
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 105 of 144
Columbus City Schools
12/1/13






Students use the incorrect values for a, b, and c in the quadratic formula, because they do
not put the equations in standard form first.
Students do not understand how to complete the square for a perfect square trinomial.
Students make mistakes with rational numbers.
Students incorrectly multiply polynomials; they believe they can just distribute the
exponent through the binomial, or when multiplying two different polynomials they
forget to multiply the inside terms.
Students make mistakes when finding the conjugate of a complex number; they often
multiply by the same binomial instead, but still cancel out the middle terms, therefore
they get the wrong sign on the last term.
Students do not recognize the pattern with imaginary numbers.
Instructional Strategies:
A – REI 1
1) Solving quadratic functions
http://www.shmoop.com/common-core-standards/ccss-hs-a-rei-1.html
This site provides written explanations and a practice sheet for solving quadratic functions.
A – REI 4
1) Throwing an Interception:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadequ_tn
_062213.pdf
In this lesson students develop the quadratic formula to determine the x-intercepts of the
function.
A – REI 4a
1) Completing the Square
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20By%20Groupi
ng.pdf
This website provides practice for completing the square.
2) Students will complete the “Sorting Activity” (included in this Curriculum Guide). In this
activity students will look at the different ways a quadratic equation can be represented (e.g.
vertex form, trinomial, factored, vertex form, as a parabola). This activity can be used as a
precursor to the “Completing the Square” activity. It is important at the conclusion of the
activity to emphasize that completing the square makes it easier to: (i) determine the
minimum value of a quadratic with a positive leading term; (ii) find the roots; and (iii) draw
the graph.
3) Have students complete the activity “Learning How to Complete the Square
“Completely”” (included in this Curriculum Guide). The activity is designed to emphasize
the two primary benefits for using the technique of “Completing the Square” to
simplify/solve a quadratic equation. In particular, it will allow students to (i) locate the
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 106 of 144
Columbus City Schools
12/1/13
minimum point of a quadratic curve y = x2 + bx + c, and (ii) put a quadratic polynomial into
vertex form which will aid students in finding the roots.
4) Have the students use the “Transformations and Completing the Square Notes” and
“Completing the Square and Transformations Practice” worksheets (included in this
Curriculum Guide). The students will be able to graph any quadratic using transformations.
The students will also understand how the vertex of a parabola relates to the vertex form of a
quadratic equation.
5) Students should complete “Discovery of Completing the Square” (included in this
Curriculum Guide) to be able to convert standard form quadratic equations into vertex form.
Some teacher instruction may be required in addition to this activity.
6) Give students manipulatives (i. e., Algebra Tiles or Algeblocks) to multiply, and factor
quadratic equations. Instruct students to solve quadratic equations by completing the square
using manipulatives. Use the resource: Virtual Manipulatives (NLVM: Algebra Tiles:
http://nlvm.usu.edu/en/nav/grade_g_4.html
7) Proof Without Words:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
At this site you can find an interactive geometric proof x2 + ax = (x + a/2)2 – (a/2)2.
8) Transform a Quadratic Equation by Completing the Square:
http://learnzillion.com/lessons/1240-transform-a-quadratic-equation-by-completing-thesquare
In this lesson students will learn how to transform a quadratic equation by completing the
square.
9) Transform a Quadratic Equation by Completing the Square, a=1:
http://learnzillion.com/lessons/1239-transform-a-quadratic-equation-by-completing-thesquare-a1
In this lesson students will learn how to transform a quadratic equation by completing the
square.
10) Derive the Quadratic Formula: Completing the Square:
http://learnzillion.com/lessons/268-derive-the-quadratic-formula-completing-the-square
In this lesson students will learn how to derive the quadratic formula by completing the
square.
11) Solve a Quadratic Equation: Completing the Square (1):
http://learnzillion.com/lessons/265-solve-a-quadratic-equation-completing-the-square-1
This is 1 of 2 lessons in which students will learn how to solve a quadratic equation by
completing the square. This lesson teaches you how to complete the square with a leading
coefficient of 1.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 107 of 144
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12/1/13
12) Solve a Quadratic Equation: Completing the Square (2):
http://learnzillion.com/lessons/266-solve-a-quadratic-equation-completing-the-square-2
This is 1 of 2 lessons in which students will learn how to solve a quadratic equation by
completing the square. This lesson teaches you how to complete the square with a leading
coefficient other than 1.
13) Completing the Square:
http://www.ixl.com/math/algebra-1/complete-the-square
Students are provided interactive problems to fill in the number that makes the polynomial a
perfect-square quadratic. A tutorial is provided is the solutions offered are incorrect.
14) “Factoring by Mack”:
http://alex.state.al.us/lesson_view.php?id=24082
In this lesson students will learn a strategy to factor trinomials.
15) Completing the Square:
http://education.ti.com/en/us/activity/detail?id=0DB3F0D2FA0D4F028119DB20332F99CE
In this activity students complete the square in an algebraic expression. Students will use
algebra tiles to build a geometric model of a perfect square quadratic.
16) Completing the Square Algebraically:
http://education.ti.com/en/us/activity/detail?id=F38582092FBD46FCB8F3DCEBBBA3D496
In this Nspire lesson students will complete the square algebraically to rewrite a quadratic
expression.
17) Quadratic Formula: How to Derive:
http://patrickjmt.com/deriving-the-quadratic-formula/
This site offers a tutorial on deriving the quadratic formula.
A – REI 4b
1) Curbside Rivalry:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0
62213.pdf
In this lesson (pp. 39-44), students examine how different forms of a quadratic equation can
facilitate the solving of the equations.
2) Perfecting My Quads:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0
62213.pdf
Students building fluency with solving quadratic equations in this lesson (pp. 45-52)
3) To Be Determined:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0
62213.pdf
Students focus on the discriminant and the roots that are complex in this lesson (pp. 53-59)
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 108 of 144
Columbus City Schools
12/1/13
4) My Irrational and Imaginary Friends:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0
62213.pdf
Students work with arithmetic with imaginary numbers and complex numbers in this lesson
(pp. 60-66).
5) iNumbers:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0
62213.pdf
Students practice working with arithmetic of complex numbers and pure imaginary numbers
(pp. 67 -74).
6) Quadratics:
https://www.khanacademy.org/math/algebra/quadratics
A series of links on solving quadratics through factoring, completing the square, graphing,
and the quadratic equation are provided.
7) Quadratic Functions: http://www.mcclenahan.info/sfhs/Algebra2/LectureNotes/76_Quadratic_Functions.pdf
At this site there is a lesson on determining the intercepts and minimum and maximum
points.
8) Completing the Square (easy):
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Complete%20the%20Squar
e.pdf
Students complete the square to determine the value of “c” in a trinomial expression.
9) Completing the Square (harder):
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Completing%20the%20Squ
are.pdf
Students determine the value of “c” in a trinomial by completing the square.
10) Solving Quadratic Equations with Square Roots (Easy):
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Solving%20Quadratic%20R
oots.pdf
Practice problems, on solving equations that contain square roots, can be found at this
website.
11) Quadratic Equations with Square Roots (Hard):
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Quadratic%20Equations%2
0Square%20Roots.pdf
Students practice solving quadratic equations with square roots with real and complex
solutions.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 109 of 144
Columbus City Schools
12/1/13
12) Solving Equations by Completing the Square (Hard):
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Quadratic%20Equations%2
0By%20Completing%20the%20Square.pdf
Students will solve equations using completing the square.
13) Using the Quadratic Formula (Easy):
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Quadratic%20Formula.pdf
Students will solve equations using the quadratic formula for problems with real number
solutions.
14) Using the Quadratic Formula (Harder):
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Quadratic%20Formula.pdf
Students determine real and complex solutions to quadratic functions by using the quadratic
formula.
15) Solving Equations by Completing the Square (Easy):
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Solving%20Completing%20
Square.pdf
Students will solve equations using completing the square.
16) Discriminant:
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/The%20Discriminant.pdf
Students determine the number of real and imaginary solutions by determining the value of
the discriminant.
17) Solving Quadratic Equations using the Quadratic Formula:
http://www.montereyinstitute.org/courses/Algebra1/U10L1T3_RESOURCE/index.html
A warm up, video presentation, practice and review are provided as a lesson on solving
quadratic equations using the quadratic formula.
18) Solving Quadratic Equations by Completing the Square:
http://www.montereyinstitute.org/courses/Algebra1/U10L1T2_RESOURCE/index.html
A warm up, video presentation, practice and review are provided as a lesson on solving
quadratic equations by completing the square.
19) Solving Quadratic Equations: Cutting Corners:
http://map.mathshell.org.uk/materials/lessons.php?taskid=432
Students will solve quadratics in one variable by solving quadratic equations by taking
square roots, completing the square, using the quadratic formula, and factoring.
20) Quadratic Formula:
http://patrickjmt.com/using-the-quadratic-formula/
This site offers a video tutorial on use of the quadratic formula.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 110 of 144
Columbus City Schools
12/1/13
21) Quadratic Equations – Factoring and Quadratic Formula:
http://patrickjmt.com/quadratic-equations-factoring-and-quadratic-formula/
This tutorial offers examples for solving quadratic equations using either factoring or the
quadratic formula.
22) Solving Quadratic Equations using the Quadratic Formula – Ex 1:
http://patrickjmt.com/solving-quadratic-equations-using-the-quadratic-formula-ex-1/
The tutorial offers examples for solving equations using the quadratic formula.
23) Solving Quadratic Equations using the Quadratic Formula – Ex 2:
http://patrickjmt.com/solving-quadratic-equations-using-the-quadratic-formula-ex-2/
The tutorial offers more examples for solving equations using the quadratic formula.
24) Solving Quadratic Equations using the Quadratic Formula – Ex 2:
http://patrickjmt.com/solving-quadratic-equations-using-the-quadratic-formula-ex-3/
The tutorial offers more examples for solving equations using the quadratic formula.
25) Quadratic Equations, Discriminant, Quadratic Formula:
http://www.regentsprep.org/Regents/math/algtrig/ATE3/indexATE3.htm
Lessons, practice and teacher resources are provided for solving quadratic equations using
the quadratic formula.
26) Solve a Quadratic Equation using Square Roots:
http://www.ixl.com/math/algebra-1/solve-a-quadratic-equation-using-square-roots
Students are provided problems to determine the solution to a quadratic equation by taking
square roots. A tutorial is provided is the solutions offered are incorrect.
27) Solve a Quadratic Equation by Completing the Square:
http://www.ixl.com/math/algebra-1/solve-a-quadratic-equation-by-completing-the-square
Students are provided problem to determine the solution of a quadratic equation by
completing the square. A tutorial is provided is the solutions offered are incorrect.
28) Solve a Quadratic Equation using the Quadratic Formula:
http://www.ixl.com/math/algebra-1/solve-a-quadratic-equation-using-the-quadratic-formula
Students are provided problems to determine the solution of a quadratic equation with the
quadratic formula. A tutorial is provided is the solutions offered are incorrect.
29) Using the Discriminant:
http://www.ixl.com/math/algebra-1/using-the-discriminant
Students are provided problems to determine the number of solutions for a quadratic
equation. A tutorial is provided is the solutions offered are incorrect.
30) Solve Quadratic Equations:
http://www.ixl.com/math/geometry/solve-quadratic-equations
Students are provided problems to determine the solution of a quadratic equation using
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 111 of 144
Columbus City Schools
12/1/13
different methods. A tutorial is provided is the solutions offered are incorrect.
A – SSE 3b
1) Math.A-SSE.3b:
http://www.shmoop.com/common-core-standards/ccss-hs-a-sse-3b.html
Written instructions for solving quadratic equations by completing the square can be found at
this site.
2) Completing the Square:
http://www.mathworksheetsland.com/algebra/5squareinquad/ip.pdf
Printable worksheets and lessons are provided for students to practice determining the “c” of
a perfect square trinomial.
F – IF 8
1) Use the task, “Quadratic (2009),” found at the Inside Mathematics website. Have students
demonstrate their understanding of quadratic functions given different representations.
Students will interpret rates of change given graphical and numerical data. Ask them to
identify the minimum points and determine the solutions of these functions algebraically. In
this activity students use a verbal description to create an equation in vertex form, and
expand it to standard form. Students are then asked to examine the graph, along with a
horizontal line and another linear graph. Students must identify the vertex, as well as the
intersection points for the different lines with the parabola. Students must also complete the
algebra to get the same results, and then go a step further and identify where the graph of the
parabola equals zero. http://insidemathematics.org/common-core-math-tasks/highschool/HS-A-2009%20Quadratic2009.pdf
2) Give students instructions on creating a graphic organizer. Instruct them to use the organizer
to compare quadratic functions using the process of factoring, completing the square and
graphing. There are websites for with examples of graphic organizers. http://www.teachnology.com/worksheets/graphic/
3) Building the Perfect Square:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_0
62213.pdf
In this lesson (pp. `14-22), students use visual and algebraic approaches to completing the
square.
F – IF 8a
1) Lining Up Quadratics:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_0
62213.pdf
In this lesson (pp. 23-28), students will focus on the vertex and intercepts for quadratics.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 112 of 144
Columbus City Schools
12/1/13
2) Area "FOILed" Again!
http://education.ti.com/en/us/activity/detail?id=E0A02061CC2B4007B4EC672574B28016
Students practice finding rectangular areas with algebraic expressions for the lengths of the
sides.
3) Factor Fixin’:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_0
62213.pdf
In this lesson (pp. 29-33), students focus on connecting the factored and expanded or
standard forms of a quadratic.
4) I’ve Got a Fill-in:
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_0
62213.pdf
In this lesson (pp. 34-41), students build fluency in rewriting and connecting different forms
of a quadratic.
5) Forming Quadratics:
http://map.mathshell.org/materials/lessons.php?taskid=224
In this lesson students demonstrate their understanding of the factored form of the function
and can identify the roots of the graphs; understand how the completed square form of the
function can identify the maximum or minimum of a graph; and understand how the standard
form can provide the graphs’ intercepts.
6) Proof without Words: Completing the Square:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
This site provides an interactive geometric proof for students to understand the concept of
completing the square.
7) Practice:
http://www.ixl.com/math/algebra-1/solve-a-quadratic-equation-by-completing-the-square
Students complete the square and write their answers as integers, proper or improper
fractions in simplest form, or decimals rounded to the hundredths place.
8) Completing the Square:
http://ccssmath.org/?s=F-IF+8+quadratics
Students use algebra tiles to build a geometric model of a perfect square trinomial. They will
complete the square and recognize the characteristics of a perfect square.
A – CED 1
1) Applications of Quadratic Functions:
http://www.montereyinstitute.org/courses/Algebra1/U10L2T1_RESOURCE/index.html
A warm up, video presentation, practice and review problems are provided for creating
algebraic models for quadratic situations and solving them.
2) Math in Basketball Lesson Plan:
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 113 of 144
Columbus City Schools
12/1/13
http://www.thirteen.org/get-the-math/files/2012/08/Math-in-Basketball-Full-Lesson-FINAL8.16.12.pdf
Using video segments and interactive on the web student explore quadratic functions. This
site offers the lesson plan, student activity sheets and answer keys.
N – CN 1
1) Determine whether a square root is real or imaginary:
http://learnzillion.com/lessons/225-determine-whether-a-square-root-is-real-or-imaginary
In this lesson students will learn how to determine whether a square root is real or imaginary.
2) Classifying Complex Numbers:
http://alex.state.al.us/lesson_view.php?id=11364
This lesson has 4 activities. The first section is a teacher Power Point presentation of the
relationship between the sets of complex, real and imaginary numbers. The class group or
individual activity is a Power Point lesson where students select the appropriate set (strictly
complex number, strictly real numbers and strictly imaginary numbers) by clicking on it. A
second activity asks students to classify complex numbers into subsets of strictly complex,
strictly real or strictly imaginary. The fourth section is a test.
3) Write the square root of negative number as imaginary:
http://learnzillion.com/lessons/226-write-the-sq-root-of-neg-number-as-imaginary
In this lesson you will learn how to write the square root of a negative number as imaginary.
4) Classify complex numbers as real or imaginary:
http://learnzillion.com/lessons/227-classify-complex-numbers-as-real-or-imaginary
In this lesson you will learn how to classify complex numbers as real or imaginary.
N – CN 2
1) Complex Number Addition:
http://education.ti.com/en/us/activity/detail?id=07EF321269B64BB398EABD1C0E0D9061
This lesson involves the addition of two complex numbers. Students compute the sum of two
complex numbers and visually and geometrically describe the sum.
2) Complex Numbers:
http://www.regentsprep.org/Regents/math/algtrig/ATO6/ImagineLes.htm
This website provides lessons, practice and teacher resources for complex numbers.
3) Operations with Complex Numbers:
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Operations%20with%20Co
mplex%20Numbers.pdf
Students practice simplifying complex numbers with this assignment.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
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Columbus City Schools
12/1/13
4) Properties of Complex Numbers:
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Properties%20of%20Compl
ex%20Numbers.pdf
Students determine the absolute value of complex numbers and graph complex numbers with
this assignment.
5) Rationalizing Imaginary Denominators:
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Rationalizing%20Imaginary
%20Denominators.pdf
Students simplify expressions by rationalizing the denominators.
6) Adding and Subtracting Complex Numbers:
http://www.regentsprep.org/Regents/math/algtrig/ATO6/lessonadd.htm
This lesson provides instructions and practice problems to add and subtract complex
numbers.
7) Complex Number Multiplication:
http://education.ti.com/en/us/activity/detail?id=95F04720818B470A9B87DB4E44A23E44
This lesson involves the product of complex numbers, powers of i and complex conjugates.
8) Complex Numbers:
http://education.ti.com/en/us/activity/detail?id=6FD90593B6FF446CB9BE76C9AF380ECE
Students calculate problems from the student worksheet to determine the rules for adding,
subtracting, multiplying, and dividing complex numbers.
9) Multiplying and Dividing Complex Numbers:
http://www.regentsprep.org/Regents/math/algtrig/ATO6/multlesson.htm
This lesson provides instructions and practice problems to multiply and divide complex
numbers.
10) Practice with Arithmetic of Complex Numbers:
http://www.regentsprep.org/Regents/math/algtrig/ATO6/practicepageadd.htm
Practice is provided on adding and subtracting complex numbers.
11) Practice with Multiplying and Dividing Complex Numbers:
http://www.regentsprep.org/Regents/math/algtrig/ATO6/multprac.htm
Practice is provided on multiplying and dividing complex numbers.
N – CN 7
1) Complex Numbers and the Quadratic Formula:
http://www.purplemath.com/modules/complex3.htm
This site provides a written description of how to use the quadratic formula to determine
imaginary solutions.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
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Columbus City Schools
12/1/13
2) Solving Quadratic Equations with Complex Roots:
http://www.regentsprep.org/Regents/math/algtrig/ATE3/quadcomlesson.htm
The lesson provides notes and examples on solving quadratic equations with complex roots.
3) Practice Solving Quadratic Equations with Complex Roots:
http://www.regentsprep.org/Regents/math/algtrig/ATE3/quadcompractice.htm
A set of practice problems are provided at this site.
4) Complex Numbers Introduction:
http://www.purplemath.com/modules/complex.htm
A video presentation and notes are provided on complex numbers.
5) Complex Roots from the Quadratic Formula:
http://www.youtube.com/watch?feature=player_embedded&v=dnjK4DPqh0k
A Khan Academy video presentation of determining complex roots is provided at this site.
6) Determine whether a number is real or imaginary: http://learnzillion.com/lessons/228determine-whether-a-number-is-real-or-imaginary-isolating-the-quadratic-term
In this lesson students will learn how to determine whether a number is real or imaginary by
isolating the quadratic term.
7) Solve quadratic equations with real coefficients: http://learnzillion.com/lessons/230solve-quadratic-equations-with-real-coefficients-using-the-quadratic-formula
In this lesson you will learn how to solve quadratic equations with real coefficients by using
the quadratic formula.
8) Determine whether a number is real or imaginary:
http://learnzillion.com/lessons/229-determine-whether-a-number-is-real-or-imaginarycalculating-the-value-of-the-discriminant
In this lesson you will learn how to determine whether a number is real or imaginary by
calculating the value of the discriminant.
9) Solve equations:
http://learnzillion.com/lessons/231-solve-equations-completing-the-square
In this lesson you will learn how to solve equations by completing the square.
Reteach:
1) Forming Quadratics:
http://map.mathshell.org/materials/download.php?fileid=700
In this lesson, students will work with different algebraic forms of a quadratic function to
understand the properties of different representations (graphical). Students will identify
roots
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 116 of 144
Columbus City Schools
12/1/13
by factoring the quadratic equations, complete the square to determine minimum or
maximum points and use the standard form of the equation to find the y-intercept.
2) Completing the Square:
http://advancedalgebra.gcs.hs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=445175&fid
=1732523&sessionid=
This is a re-teach practice sheet with an answer key provided for students to solve
quadratic equations using completing the square.
3) Quadratic Formula:
http://advancedalgebra.gcs.hs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=445175&fid
=1732543&sessionid=
This is a re-teach practice sheet with an answer key provided for students to solve quadratic
equations with the quadratic formula.
Extensions:
1) Horseshoes in Flight:
http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSMTask/Horseshoes.pdf
The height of the thrown horseshoe depends on the time that has passed since it was
released. Students will derive information about the flight of a horseshoe from the graph
and the four given equivalent algebraic expressions that describe its flight and complete
the activity sheet.
2) Bridging the Gap:
http://www.oame.on.ca/main/files/OMCA%20MCF3M/Unit%204%20Midterm%20SP%2
0Task.pdf
This midterm summative performace task has a series of lessons in which students will:
solve a problem by creating a scale model, collect data and create an algebraic model;
demonstrate their understanding of connections between numeric, graphical, and algebraic
representations of quadratic functions; and solve real-world problems.
3) Performance Task: http://insidemathematics.org/commoncore-math-tasks/high-school/HS-A2009%20Quadratic2009.pdf
Students will work with a quadratic function in various forms.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 117 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Sorting Activity
Work with a partner to cut and sort the cards. Label a column in your notes so that it looks like
the column on the left-hand margin of this page. Tape or glue two sets of six cards each in your
notes.
Quadratic
Equation
y = x2 + 5x + 6
y = x2 + 5x – 6
y = (x + 2)(x + 3)
y = (x – 2)(x – 3)
Factored
2
Completing the
Square/Vertex Form
Minimum Point
2
5  49

y x  
2
4

 5 49 
Minimum at  - , 
4 
 2
Solutions
y = 0 when x = -
5 1

2 2
5 1

y x  
2 4

 5 1
Minimum at  - , - 
 2 4
y = 0 when x = -
5 7

2 2
Graph
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 118 of 144
Columbus City Schools
12/1/13
Quadratic
Equation
y = x2 – 5x + 6
y = x2 – 5x – 6
y = (x + 1)(x – 6)
y = (x – 1)(x + 6)
Factored
2
Completing the
Square/Vertex
Form
2
5  49

y x  
2
4

5 1

y x  
2 4

5 1
Minimum at  , - 
2 4
 5 49 
Minimum at  , 
4 
2
Minimum Point
y = 0 when x =
5 7

2 2
y = 0 when x =
5 1

2 2
Solutions
Graphs
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 119 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Sorting Activity
Answer Key
The answers students should have are aligned in the two columns.
Quadratic
Equation
y = x2 - 5x + 6
y = x2 + 5x + 6
Factored
y = (x - 2)(x - 3)
y = (x + 2)(x + 3)
2
Completing
the
Square/Vertex
Form
Minimum
Point
Solutions
2
5 1

y x  
2 4

5 1

y x  
2 4

5 1
Minimum at  , - 
2 4
 5 1
Minimum at  - , - 
 2 4
y = 0 when x =
5 1

2 2
y = 0 when x = -
5 1

2 2
Graphs
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 120 of 144
Columbus City Schools
12/1/13
Quadratic
Equation
y = x2 + 5x – 6
y = x2 – 5x – 6
y = (x – 1)(x + 6)
y = (x + 1)(x – 6)
Factored
Completing
the
Square/Vertex
Form
Minimum
Point
Solutions
2
2
5  49

y x  
2
4

5  49

y x  
2
4

 5 49 
Minimum at  - , 
4 
 2
 5 49 
Minimum at  , 
4 
2
y = 0 when x = -
5 7

2 2
y = 0 when x =
5 7

2 2
Graphs
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 121 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Learning How to Complete the Square “Completely”
1. In words, explain what is the same and what is different about the equations (x – 2)2 = 25 and
x2 – 4x + 4 = 25.
2. x2 – 4x + 4 is called what kind of trinomial? Answer: _______________________________
3. (x – 2)2 is called what kind of binomial?
Answer: _______________________________
4. List a few things you can say about the graph of y = (x – 2)2 .
5. Will (x – 2)2 ever be negative? Explain.
6. Will (x + 2)2 ever be negative? Explain.
7. List a few things about the graph of y = (x – 2)2 + 3.
8. What is the minimum value of y = (x – 2)2 – 10?
Answer: _______________
What is the minimum value of y = (x + 2)2 + 10?
Answer: _______________
9. If y = (x – 2)2 + 3, then y is a quadratic in _____________________ form.
10. The vertex of the parabola y = (x – 2)2 + 3 is ___________________ .
11. The vertex of the parabola y = (x + 3)2 – 2 is ___________________ .
12. What is the location of the minimum point for y = (x + 3)2 – 2 ? Answer: ______________
13. Completing the square allows us to write a quadratic in ___________________ form by
changing the given trinomial into a _____________________________ trinomial and then
factoring it into a binomial-squared.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 122 of 144
Columbus City Schools
12/1/13
For problems 14-17, write the quadratic in vertex form by completing the square and give the
minimum value.
14.
y = x2 + 6x – 11
15.
y = x2 – 10x + 16
Vertex form:___________________
Vertex form:__________________
Minimum: ___________________
Minimum: __________________
16.
y = x2+2x – 8
17.
y = x2 + 5x + 6
Vertex form:___________________
Vertex form:__________________
Minimum: ___________________
Minimum: __________________
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 123 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Learning How to Complete the Square “Completely”
Answer Key
1. In words, explain what is the same and what is different about the equations (x – 2)2 = 25 and
x2 – 4x + 4 = 25.
The equations look different, but algebraically they are the same because
(x – 2)2 = x2 – 4x + 4 when multiplied out. Also, the roots are the same.
2. x2 – 4x + 4 is called what kind of trinomial? Answer:__perfect square trinomial_________
3. (x – 2)2 is called what kind of binomial?
Answer:__binomial squared_______________
4. List a few things you can say about the graph of y = (x – 2)2 .
(i) it is a parabola
(iii) minimum value at y = 0
(v) when x = 2, then y = 0 (e.g. x-intercept = 2)
(ii) it has the shape of y = x2
(iv) it is symmetric about the line x = 2
5. Will (x – 2)2 ever be negative? Explain.
No, because the whole binomial is being squared and a ‘squared’ number will always be
positive.
6. Will (x + 2)2 ever be negative? Explain.
No, because the whole binomial is being squared and a ‘squared’ number will always be
positive.
7. List a few things about the graph of y = (x – 2)2 + 3.
The minimum value occurs at y = 3. It is a horizontal shift 2 to the right and a vertical
shift 3-up of the graph of y = x2.
8. What is the minimum value of y = (x – 2)2 – 10?
What is the minimum value of y = (x + 2)2 + 10?
Answer: -10
Answer: -10
9. If y = (x – 2)2 + 3, then y is a quadratic in ____vertex______ form.
10. The vertex of the parabola y = (x – 2)2 + 3 is (2,3) .
11. The vertex of the parabola y = (x + 3)2 – 2 is (-3, -2) .
12. What is the location of the minimum point for y = (x + 3)2 – 2 ? Answer: (-3, -2)
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 124 of 144
Columbus City Schools
12/1/13
13. Completing the square allows us to write a quadratic in __vertex ______ form by
changing the given trinomial into a ____perfect-square __________ trinomial and then
factoring it into a binomial-squared.
For problems 14-17, write the quadratic in vertex form by completing the square and give the
minimum value.
14.
y = x2 + 6x – 11
15.
y = x2 – 10x + 16
Vertex form: y = (x + 3)2 – 20
Vertex form: y = (x – 5)2 – 9
Minimum: -20
Minimum: -9
16.
y = x2+2x – 8
17.
y = x2 + 5x + 6
Vertex form: y = (x + 1)2 – 9
Vertex form: y = (x + 5/2)2 – ¼__
Minimum: 9
Minimum: -1/4
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 125 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Transformations and Completing the Square Notes
The function f  x   x 2 is the parent function of all quadratics. Every quadratic can be
transformed from this graph. By using completing the square, all quadratics can be rewritten, in
what some people call, the vertex form for a quadratic equation.
2
The vertex form is: y  a  x  c   d . Think of this as a template.
Transformations:
a
If a  0 , then the graph is reflected about the x-axis.
If a  1 , then a vertical stretch by a factor of a occurs.
If 0  a  1 , then a vertical shrink by a factor of a occurs.
c
If c  0 , then the graph will shift c units to the right.
If c  0 , then the graph will shift c units to the left.
d
If d  0 , then the graph will shift down d units.
If d  0 , then the graph will shift up d units.
So, if y  - 2  x  3  1 , the following transformations would occur to the graph of f  x   x 2 .
2
1) Reflection about the x-axis, since a  - 2 . (The y-coordinates will become opposites.)
2) Vertical stretch by a factor of 2, because - 2  2 . (Multiply the y-coordinates by 2.)
3) Horizontal shift 3 units to the right, because c = 3. (Add 3 to the x-coordinates.)
4) Vertical shift up 1 unit, since d  1 . (Add 1 to the y-coordinates.)
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 126 of 144
Columbus City Schools
12/1/13
Below you will see the transformations that are applied to f(x) = x2.
The graph of f(x) = x2.
A reflection about the x-axis.
A vertical stretch by a factor of 2.
Horizontal shift to the right 3 units.
A vertical shift up 1 unit.
What is the value of c?
What is the value of d?
What are the coordinates of the vertex for
the parabola on the left?
Is there a connection between the vertex and
the value of c and d?
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 127 of 144
Columbus City Schools
12/1/13
Practice:
State the transformations that would occur to f(x) = x2 and the coordinates of the vertex for each
of the new graphs. Using the transformations and the points (-2,4), (-1,1), (0,0), (1,1), and (2,4)
from the graph of f(x) = x2, find the coordinates of the transformed points and graph the new
function. Show the mapping of the points.
1. y = 3(x – 1) 2 – 5
2. y  
2
2
 x  5  7
3
3. a) State the transformations that would occur to f(x) = x2, if y = - 3(x + 4) 2 + 2.
b) Using the transformations stated in part a and the point (2,4) from the graph of y = x2,
give the coordinates for the transformed point.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 128 of 144
Columbus City Schools
12/1/13
How to complete the square
Let
Group the quadratic and linear term together.
If the leading coefficient is not 1, factor it out.
y  x2  4 x  5
y   x2  4x   5


y  x2  4x   - 2  5   - 2
2
2
You will need to create a perfect trinomial
square (take half of the linear term and square
it). But because you do not want to change the
equivalence of the equation; you will need to
add and subtract the same number to the one
side of the equation.
y   x  2  5  4   x  2  1
Rewrite the perfect trinomial square in its
factored form as a square of a binomial.
Thus y  x2  4x  5 =  x  2   1
Simplify the constants.
2
2
2
State the transformations. What are the
coordinates of the vertex?
Completing the Square, when the leading coefficient is not 1.
Group the quadratic and linear term together.
2
y  -2x  6x  7
y   - 2x2  6x   7
Factor the leading coefficient out of the linear
and quadratic terms.
y  - 2  x 2  3x   7
2
2

 3 
 3
y  - 2  x 2  3x   -    7   - 2   - 

 2  
 2

3


 9 
y  - 2  x    7   - 2    
2

 4 

2
2
Create a perfect trinomial square but remember
to subtract the exact same number from the
constant. Remember you have a multiplier in
front that will need to be included when
subtracting.
Rewrite the perfect trinomial square in it’s
factored form: square of a binomial.
2
3
3 5

 9

y  -2  x    7   -   -2  x   
2
2 2

 2

Simplify the constant.
State the transformations. What are the
coordinates for the vertex?
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 129 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Transformations and Completing the Square Notes
Answer Key
The function f  x   x is the parent function of all quadratics. Every quadratic can be
transformed from this graph. By using completing the square, all quadratics can be rewritten, in
what some people call, the vertex form for a quadratic equation.
2
The vertex form is: y  a  x  c   d . Think of this as a template.
2
Transformations:
a
If a  0 , then the graph is reflected about the x-axis.
If a  1 , then a vertical stretch by a factor of a occurs.
If 0  a  1 , then a vertical shrink by a factor of a occurs.
c
If c  0 , then the graph will shift c units to the right.
If c  0 , then the graph will shift c units to the left.
d
If d  0 , then the graph will shift down d units.
If d  0 , then the graph will shift up d units.
So, if y  - 2  x  3  1 , the following transformations would occur to the graph of f  x   x 2 .
2
1) Reflection about the x-axis, since a  - 2 . (The y-coordinates will become opposites.)
2) Vertical stretch by a factor of 2, because - 2  2 . (Multiply the y-coordinates by 2.)
3) Horizontal shift 3 units to the right, because c = 3. (Add 3 to the x-coordinates.)
4) Vertical shift up 1 unit, since d  1 . (Add 1 to the y-coordinates.)
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 130 of 144
Columbus City Schools
12/1/13
Below you will see the transformations that are applied to f(x) = x2.
The graph of f(x) = x2.
A reflection about the x-axis.
A vertical stretch by a factor of 2.
Horizontal shift to the right 3 units.
A vertical shift up 1 unit.
What is the value of c?
3
What is the value of d?
1
What are the coordinates of the vertex for
the parabola on the left?
(3,1)
Is there a connection between the vertex and
the value of c and d?
Yes there is a connection. The
coordinates for the vertex correspond to
the c and d values. The vertex can be
written as (c,d).
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 131 of 144
Columbus City Schools
12/1/13
Practice:
State the transformations that would occur to f(x) = x2 and the coordinates of the vertex for each
of the new graphs. Using the transformations and the points (-2,4), (-1,1), (0,0), (1,1), and (2,4)
from the graph of f(x) = x2, find the coordinates of the transformed points and graph the new
function. Show the mapping of the points.
1. y = 3(x – 1) 2 – 5
A vertical stretch by a factor of 3.
A horizontal shift to the right 1 unit.
A vertical shift down 5 units.
The coordinates of the vertex are (1, - 5).
 - 2, 4    - 2, 12    -1, 12    -1, 7 
 -1, 1   -1, 3    0, 3    0, - 2 
 0, 0    0, 0   1, 0   1, - 5 
1, 1  1, 3    2, 3    2, - 2 
 2, 4    2, 12    3, 12    3, 7 
2
2
 x  5  7
3
A reflection about the x-axis.
2. y  -
A vertical shrink by a factor of
2
.
3
A horizontal shift left 5 units.
A vertical shift up 7 units.
The coordinates of the vertex are (- 5, 7)
 - 2, 4    - 2, - 4    - 2, - 2.67    - 7, - 2.67    - 7, 4.33 
 -1, 1   -1, -1   -1, - 0.67    - 6, - 0.67    - 6, 6.33 
 0, 0    0, 0    0, 0    - 5, 0    - 5, 7 
1, 1  1, -1  1, - 0.67    - 4, - 0.67    - 4, 6.33 
 2, 4    2, - 4    2, - 2.67    - 3, - 2.67    - 3, 4.33 
3.
a) State the transformations that would occur to f(x) = x2, if y = - 3(x + 4)2 + 2.
A reflection about the x-axis.
A vertical stretch by a factor of 3.
A horizontal shift to the left 4.
A vertical shift up 2 units.
b) Using the transformations stated in part a and the point (2,4) from the graph of y = x2,
give the coordinates for the transformed point.
(2, 4)  (2, - 4)  (2, - 12)  (- 2, - 12)  (- 2, - 10)
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 132 of 144
Columbus City Schools
12/1/13
How to complete the square
y  x2  4 x  5
Let
Group the quadratic and linear term together.
y   x2  4x   5

If the leading coefficient is not 1, factor it out.

y  x2  4x   - 2  5   - 2
2
2
y   x  2  5  4   x  2  1
You will need to create a perfect trinomial
square (take half of the linear term and square
it). But because you do not want to change the
equivalence of the equation; you will need to
add and subtract the same number to the one
side of the equation.
Thus y  x2  4x  5 =  x  2   1
Rewrite the perfect trinomial square in its
factored form as a square of a binomial.
2
2
2
Simplify the constants.
Shift left 2 and up 1.
Vertex (2, 1)
Completing the Square, when the leading coefficient is different from 1.
y  -2x2  6x  7
State the
transformations,
and
find
the vertex.
Group
the quadratic
and
linear
term
y   - 2x2  6x   7
together.
y  - 2  x 2  3x   7
Factor the leading coefficient out of the
linear and quadratic terms.
2
2

 3 
 3
y  - 2  x 2  3x   -    7   - 2   - 

 2  
 2

3


 9 
y  - 2  x    7   - 2    
2

 4 

2
2
2
3
3 5

 9

y  -2  x    7   -   -2  x   
2
2 2

 2

Create a perfect trinomial square but
remember to subtract the exact same
number from the constant. Remember
you have a multiplier in front that will
need to be included when subtracting.
Rewrite the perfect trinomial square in it’s
factored form: square of a binomial.
Simplify the constant.
State
are and
the
Reflect about the x-axis, a vertical stretch by a factor
ofthe
2, atransformations.
horizontal shiftWhat
left one
one-half units, and a vertical shift down 2 and one-half
units. for
Thethe
coordinates
coordinates
vertex? for the
3
5


vertex are  , -  .
2 2
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 133 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Completing the Square and Transformations Practice
Complete the square for each quadratic equation, state the transformations, show the
transformation of the points (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), from the graph of y = x2, give the
new coordinates of the vertex, and sketch the new graph.
1. y = x2 + 4x – 3
2. y = 3x2 – 6x + 7
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 134 of 144
Columbus City Schools
12/1/13
3. y 
1 2
x  x4
2
4. y = - 2x2 + 4x + 1
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 135 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Completing the Square and Transformations Practice
Answer Key
Complete the square for each quadratic equation, state the transformations, show the
transformation of the points (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), from the graph of y = x2,give the
new coordinates of the vertex, and sketch the new graph using the transformed points.
1. y = x2 + 4x – 3
y = (x2 + 4x) – 3
y = (x2 + 4x + 4) – 3 – 4
y = (x + 2)2 – 7
(-2, 4)
(- 1, 1)
(0, 0)
(1, 1)
(2, 4)
 (- 4, 4)
 (- 3, 1)
 (- 2, 0)
 (- 1, 1)
 (0, 4)
 (- 4, - 3)
 (- 3, - 6)
 (- 2, - 7)
 (- 1,- 6)
 (0, - 3)
The transformations are:
A horizontal shift to the left 2 units.
A vertical shift down 7 units.
The coordinates of the vertex are (- 2, - 7).
2. y = 3x2 – 6x + 7
y = 3(x2 – 2x) + 7
y = 3(x2 – 2x + 1) + 7 – 3
y = 3(x – 1)2 + 4
(- 2, 4)
(- 1, 1)
(0, 0)
(1, 1)
(2, 4)
 (- 2, 12)
 (- 1, 3)
 (0, 0)
 (1, 3)
 (2, 12)
 (- 1, 12)  (- 1, 16)
 (0, 3)  (0, 7)
 (1, 0)  (1, 4)
 (2, 3)  (2, 7)
 (3, 12)  (3, 16)
The transformations are:
A vertical stretch by a factor of 3.
A horizontal shift right 1 unit.
A vertical shift up 4 units.
The coordinates of the vertex are (1, 4).
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 136 of 144
Columbus City Schools
12/1/13
3. y 
1 2
x  x4
2
1 2
 x  2x  4
2
1
1
y   x 2  2 x  1  4 
2
2
1
1
2
y   x  1  4
2
2
The transformations are:
y
1
.
2
A horizontal shift to the left 1 unit.
1
A vertical shift down 4 units.
2
1

(- 2, 4)  (- 2, 2)  (- 3, 2)   - 3, - 2 
2

1
 1 
(- 1, 1)   - 1,    - 2,   (- 2, - 4)
2
 2 
1

(0, 0)  (0, 0)  (- 1, 0)   - 1, - 4 
2

 1  1
(1, 1)   1,    0,   (0, - 4)
 2  2
1

(2, 4)  (2, 2)  (1, 2)   1, - 2 
2

1

The coordinates of the vertex are  - 1, - 4  .
2

2
4. y = - 2x + 4x + 1
y = - 2(x2 - 2x) + 1
y = - 2(x2 - 2x + 1) + 1 + 2
y = - 2(x - 1)2 + 3
The transformations are:
A reflection about the x-axis.
A vertical stretch by a factor of 2.
A horizontal shift to the right 1 unit.
A vertical shift up 3 units.
(- 2, 4)  (- 2, - 4)  (- 2, - 8)  (- 1, - 8)  (- 1, - 5)
(- 1, 1)  (- 1, - 1)  (- 1, - 2)  (0, - 2)  (0, 1)
(0, 0)  (0, 0)  (0, 0)  (1, 0)  (1, 3)
(1, 1)  (1, - 1)  (1, - 2)  (2, - 2)  (2, 1)
(2, 4)  (2, - 4)  (2, - 8)  (3, - 8)  (3, - 5)
The coordinates of the vertex are (1, 3)
A vertical shrink by a factor of
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 137 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Discovery of Completing the Square
Multiply the following binomial expressions and simplify.
1. (x + 4)(x + 4)
2. (a – 3)(a – 3)
SHOW WORK
______ + ______ + ______ + ______
SIMPLIFY
SHOW WORK
______ + ______ + ______ + ______
SIMPLIFY
______ + ______ + ______
______ + ______ + ______
3. (y – 1)(y – 1)
4. (w + 5)(w + 5)
SHOW WORK
SHOW WORK
______ + ______ + ______ + ______
SIMPLIFY
______ + ______ + ______ + ______
SIMPLIFY
______ + ______ + ______
______ + ______ + ______
5. What do you notice about all of the above problems?
6. Describe how the final result compares to the original problem? Be specific.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 138 of 144
Columbus City Schools
12/1/13
Factor the following trinomials. In other words, do the reverse of #1-4.
7. x2 + 4x + 4
8. r2 – 12r + 36
9. p2 + 20p + 100
10. q2 – 6q + 9
Fill in the missing number to make the following problems perfect square trinomials.
11. x2 + 14x + _______
12. x2 – 18x + _______
13. y2 + _______ + 16
14. t2 + _______ + 25
15. m2 – 3m + ______
16. k2 + 9k + _______
Each of the following problems is not a perfect square trinomial. Your job is to turn them into
perfect squares by “completing the square.” Answer i) – iv) to help “complete the square” for
each problem.
17. b2 +16b +30 = 0
i) In order for this to be a perfect square, the constant term should be __________.
ii) I need to add __________ to the left side of the equation to make it a perfect square
trinomial, but I also must add __________ to the right side of the equation to keep it
balanced. Show this work.
iii) Factor the left side into a perfect square. Show this work.
iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this
work.
v) The vertex of this equation is ________________.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 139 of 144
Columbus City Schools
12/1/13
18. x2 + 4x + 7=0
i) In order for this to be a perfect square, the constant term should be __________.
ii) I need to add __________ to the left side of the equation to make it a perfect square
trinomial, but I also must add __________ to the right side of the equation to keep it
balanced. Show this work.
iii) Factor the left side into a perfect square. Show this work.
iv) Bring the constant back over to the left side to set the equation equal to zero again. Show
this work.
v) The vertex of this equation is ________________.
19. f2 – 6f + 5 = 0
i) In order for this to be a perfect square, the constant term should be __________.
ii) I need to add __________ to the left side of the equation to make it a perfect square
trinomial, but I also must add __________ to the right side of the equation to keep it balanced.
Show this work.
iii) Factor the left side into a perfect square. Show this work.
iv) Bring the constant back over to the left side to set the equation equal to zero again. Show
this work.
v) The vertex of this equation is ________________.
20. r2 – 10r – 4 = 0
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 140 of 144
Columbus City Schools
12/1/13
i) In order for this to be a perfect square, the constant term should be __________.
ii) I need to add __________ to the left side of the equation to make it a perfect square
trinomial, but I also must add __________ to the right side of the equation to keep it balanced.
Show this work.
iii) Factor the left side into a perfect square. Show this work.
iv) Bring the constant back over to the left side to set the equation equal to zero again. Show
this work.
v) The vertex of this equation is _______________.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 141 of 144
Columbus City Schools
12/1/13
A – REI 4a
Name_______________________________________________ Date___________ Period_____
Discovery of Completing the Square
Answer Key
Multiply the following binomial expressions and simplify.
1. (x + 4)(x + 4)
2. (a – 3)(a – 3)
SHOW WORK
x2
+
4x
SHOW WORK
+
4x +
16
a2
+ -3a
+
-3a
+
9
SIMPLIFY
SIMPLIFY
__x2__ + __8x__ + __16__
__a2__ + __-6a__ + __9__
3. (y – 1)(y – 1)
4. (w + 5)(w + 5)
SHOW WORK
SHOW WORK
__y2__ + __-y__ + __-y___ + __1___
__w2__ + __5w__ + __5w_ + __25__
SIMPLIFY
SIMPLIFY
__y2__ + __-2y__ + __1__
__w2__ + _10w__ + __25__
5. What do you notice about all of the above problems?
#1-#4 are all problems that have a binomial multiplied by the same binomial.
6. Describe how the final result compares to the original problem? Be specific.
The middle term is two times the number in the original problem. The last term is the
square of the number in the original problem.
Factor the following trinomials. In other words, do the reverse of #1-4.
7. x2 + 4x + 4
8. r2 – 12r + 36
(x + 2)(x + 2)
(r – 6)(r – 6)
9. p2 + 20p + 100
(p + 10)(p + 10)
10. q2 – 6q + 9
(q – 3)(q – 3)
Fill in the missing number to make the following problems perfect square trinomials.
11. x2 + 14x + __49___
12. x2 – 18x + __81___
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 142 of 144
Columbus City Schools
12/1/13
13. y2 + __8y__ + 16
14. t2 + __10t__ + 25
15. m2 – 3m + _2.25_
16. k2 + 9k + __20.25 _
Each of the following problems is not a perfect square trinomial. Your job is to turn them into
perfect squares by “completing the square.” Answer i) – iv) to help “complete the square” for
each problem.
17. b2 +16b +30 = 0
i) In order for this to be a perfect square, the constant term should be ____64____.
ii) I need to add __ 34____ to the left side of the equation to make it a perfect square trinomial,
but I also must add ___34____ to the right side of the equation to keep it balanced. Show this
work.
b2 + 16b +30 + 34 = 34
b2 + 16b + 64 = 34
iii) Factor the left side into a perfect square. Show this work.
(b + 8)2 = 34
iv) Bring the constant back over to the left side to set the equation equal to zero again. Show
this work.
(b + 8)2 – 34 = 0
v) The vertex of this equation is __(-8, -34) _.
18. x2 + 4x + 7=0
i) In order for this to be a perfect square, the constant term should be ___ 4_____.
ii) I need to add __ -3____ to the left side of the equation to make it a perfect square trinomial,
but I also must add ___ -3____ to the right side of the equation to keep it balanced. Show this
work.
x2 + 4x + 7 – 3 = -3
x2 + 4x + 4 = -3
iii) Factor the left side into a perfect square. Show this work.
(x + 2)2 = -3
iv) Bring the constant back over to the left side to set the equation equal to zero again. Show
this work.
(x + 2)2 + 3 = 0
v) The vertex of this equation is ___ (-2, 3)______.
19. f2 – 6f + 5 = 0
i) In order for this to be a perfect square, the constant term should be ___ 9_____.
ii) I need to add ____4_____ to the left side of the equation to make it a perfect square
trinomial, but I also must add ____4____ to the right side of the equation to keep it balanced.
Show this work.
f2 – 6f + 5 + 4 = 4
f2 – 6f + 9 = 4
iii) Factor the left side into a perfect square. Show this work.
(f – 3)2 = 4
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 143 of 144
Columbus City Schools
12/1/13
iv) Bring the constant back over to the left side to set the equation equal to zero again. Show
this work.
(f – 3)2 – 4 = 0
v) The vertex of this equation is ____(3, -4)____.
20. r2 – 10r – 4 = 0
i) In order for this to be a perfect square, the constant term should be ___ 25______.
ii) I need to add ___ 29____ to the left side of the equation to make it a perfect square trinomial,
but I also must add ___ 29____ to the right side of the equation to keep it balanced. Show this
work.
r2 – 10r – 4 + 29 = 29
r2 – 10r + 25 = 29
iii) Factor the left side into a perfect square. Show this work.
(r – 5)2 = 29
iv) Bring the constant back over to the left side to set the equation equal to zero again. Show
this work.
(r – 5)2 – 29 = 0
v) The vertex of this equation is __ (5, -29)___.
CCSSM II
Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a,
4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7
Quarter 3
Page 144 of 144
Columbus City Schools
12/1/13