Keansburg School District
Curriculum Management System
Believe, Understand, and Realize Goals
Mathematics: Algebra I - College and Career Ready (CCR)
Board Approved:
1
Keansburg Public Schools
Board of Education
Mrs. Judy Ferraro, President
Ms. Kimberly Kelaher-Moran, Vice President
Ms. Delores A. Bartram
Ms. Ann Marie Best
Ms. Christine Blum
Ms. Ann Commarato
Mr. Michael Donaldson
Ms. Patricia Frizell
Mr. Robert Ketch
District Administration
Mr. Gerald North, Superintendent
Dr. Thomas W. Tramaglini, Director of Curriculum, Instruction, & Funding
Ms. Michelle Derpich, Secondary Supervisor of Curriculum & Instruction
Mrs. Donna Glomb, Elementary Supervisor of Curriculum & Instruction
Ms. Michelle Halperin-Krain, Supervisor of Data & Assessment
Dr. Brian Latwis, Supervisor of Pupil Personnel Services
Ms. Corey Lowell, Business Administrator
Jennifer Anderson
Karen Bruno
Gina Cancellieri
Giacinto Dagostino
Karen Egan
Obed Espada
Curriculum Development Committee
Maureen Hooker
Justine Ince
Tara Kukulski
Carrie Mazak
Michelle Meyers
Nicole Miragliotta
Camille Negri
Jennifer O’Keefe
Frank Reash
Roslyn Simek
2
Believe, Understand, and Realize Goals
Non-Negotiables
Graduates
that are
prepared
and
inspired
to make positive
contributions to society
2
Mission/Vision Statement
The mission of the Keansburg School District is to ensure an optimum, safe teaching and learning environment, which sets high
expectations and enables all students to reach their maximum potential. Through a joint community-wide commitment, we will
meet the diverse needs of our students and the challenges of a changing society.
Beliefs
We believe that:

All children can learn.

To meet the challenges of change, risk must be taken.

Every student is entitled to an equal educational opportunity.

It is our responsibility to enable students to succeed and become the best that they can be.

All individuals should be treated with dignity and respect.

The school system should be responsive to the diversity within our total population.

The degree of commitment and level of involvement in the decision-making processes, from the student, community, home
and school, will determine the quality of education.

Decisions should be based on the needs of the students.

Achievement will rise to the level of expectation.

Students should be taught how to learn.

The educational process should be a coordinated system of services and programs.
Curriculum Philosophy
The curriculum philosophy of the Keansburg School District is progressive. We embrace the high expectations of our students and
community towards success in the 21st Century and beyond. At the center of this ideal, we believe that all of our students can be successful.
The following are our core beliefs for all curricula:
All district curricula:

Balances policy driven trends of centralization and standardization with research and what we know is good for our students.

Balances the strong emphasis on test success and curriculum standards with how and what our students must know to be successful in
our community.

Embraces the reality that our students differ in the way they learn and perform, and personalizes instruction to meet the needs of each
learner.

Are aligned to be developmentally appropriate.
3

Provides teachers the support and flexibility to be innovative and creative to meet the needs of our students.
Mathematics Goals
To deliver a curriculum that is:

Pertinent for the success of all of our students and useful for teachers in the 21st Century.

Problem-based, where students understand the importance of mathematical concepts and applications.

Socially, emotionally, and academically driven with regards to statute and code, while focusing on what is best for each of the students
in our school district to achieve successful outcomes.

Significant in the processes of growth and development, and relevant to the students.

Differentiated with regards to our students’ abilities and needs.

Embedded with teaching responsibility, respect, and the value of hard work and self-pride over time.

Designed with both content knowledge and experiences which:
o
Are aligned from one grade level to the next, with scaffolded underpinnings of similar concepts for success.
o
Engage our diverse population for positive outcomes.
o
Build and support the language of mathematics.
o
Develop educational and mathematical independence over time.
4
5
Algebra I Scope and Sequence
Concepts/Big ideas
Year
SEPT/OCT
Block
SEPT
OCT/NOV
SEPT
DEC/JAN
OCT
Concepts/Big ideas
I.
o
o
o
o
o
o
o
o
II.
o
o
o
o
o
o
o
III.
o
o
o
o
o
o
o
Simplifying Expressions
Recognizing Properties
Order of Operations
Simplifying Algebraic Expressions
Operations with Rational Numbers
Probability
Use Scientific Notation
Simplify Exponential Expressions
Perform Simple Arithmetic Operations on Polynomials
Equations and Inequalities
Solve one step equations
Solve multi-step equations
Solve equations with absolute value
Solve and graph simple inequalities
Solve and graph compound inequalities
Solve and graph inequalities with absolute value
Solve and graph inequalities with two variables
Graphing Linear Equations
Find Slope
Identifying x and y Intercepts
Write Equations in Point-Slope Form
Write Equations in Slope-Intercept Form
Graph Using x and y-intercepts
Graph Using Slope and Y-Intercept
Determine whether lines are Parallel or Perpendicular
6
JAN/FEB
OCT
IV.
o
o
o
o
Data Analysis
Display and Interpret Data
Find Measures of Central Tendency
Graph and Interpret Scatter Plots
Find and graph the Line of Best Fit
FEB/MARCH
NOV
MARCH/APRIL
NOV/DEC
APRIL/MAY
DEC
V.
o
o
o
o
o
o
VI.
o
o
o
o
VII.
o
o
o
o
Functions
Find domain and range of a function
Graph functions using a given domain
Determine whether a relation is a function
Use and identify function notation
Solve problems involving direct variation
Solve problems involving inverse variation
Factoring
Find factors of monomials
Factor using the distributive property
Factor Trinomials
Factor Perfect Squares
Systems of Equations
Solve using elimination
Solve using graphing
Solve using substitution
Solve systems of inequalities
MAY/JUNE
DEC/JAN
VIII. Quadratic and Exponential Functions
o
Graph Quadratic Equations
o
Solve Quadratic Equations by Graphing
o
Solve Quadratic Equations by Factoring
o
Solve Quadratic Equations by Completing the Square
o
Solve Quadratic Equations by using the Quadratic
Formula
o
Graph Exponential Functions
7
Keansburg School District
Curriculum Management System
Suggested Days of
Instruction
Subject/Grade/Level:
Mathematics/Algebra I
Content
Standards /
CPI / Essential
Questions
CPI:
A.APR.1
A.SSE.1
A.SSE.1A
A.SSE.3
F.LE.5
N.CN.2
N.NRN.3
N.Q.1
S.CP.7
S.IC.3
S.IC.4
S.IC.5
S.IC.6
EQ:
Why is it
critical to use
the order of
operations
when
simplifying
expressions?
In what
Timeline:
September – October (Year)
September (Block)
Topic(s):
Simplifying Expressions
Significance of Learning Goal(s): Using educated steps to solve abstract problems as they apply to real life situations
and demonstrate how properties relate to Algebra.
Specific Learning Objective(s)
Suggested Activities
Instructional Tools / Materials / Technology / Resources /
Assessments and Assessment Models
The Students Will Be Able To:
Concept(s):
I.
Simplifying Expressions
o
Recognizing
Properties
o
Use Order of
Operations
o
Simplify and
interpret Algebraic Expressions
o
Operations with
Rational Numbers
o
Probability and
Odds
o
Use Scientific
Notation
o
Simplify
exponential expressions
o
Perform arithmetic
operations on polynomials
Meets the Standard (SWBAT):

Identify mathematical
vocabulary when evaluating
expressions

Evaluate expressions using
the order of operations
Meets Standard:
Have students make up
order of operation
problems and submit
them to the class as an
exercise.
Make a target for the
board and have students
figure out the odds and
probability of hitting the
center. Discuss if
changing the distance
matters.
Graphing calculator
Probability
http://www.shodor.org/interactivate/activities/ExpProbability/
http://www.youtube.com/watch?v=D8ziFVluofw
http://www.youtube.com/watch?v=0xm1SDlnvh4&feature=related
Order of Operations
http://www.shodor.org/interactivate/activities/OrderOfOperationsFou/
Exceeds Standard:
Scientists estimate that
there are about 1020
stars in the universe. A
cubic meter of beach
sand contains about 109
grains of sand. Suppose
all of the sand from all of
the world’s beaches is
Scientific Notation
http://www.xpmath.com/forums/arcade.php?do=play&gameid=21
Exponential Expressions
8
situations do
signed
numbers model
real life?

Recognize, perform
operations, and draw conclusions
involving rational and irrational
numbers.

Design expressions that
demonstrate properties

Use the rules of probability
to compute probabilities of
compound and independent events

Make inferences and justify
conclusions from sample surveys
and experiments

Use scientific notation to
write very large or very small
numbers

Use scientific notation to
simplify expressions with very large
or small numbers

simplify expressions with
positive and negative exponents

Add, subtract, multiply and
divide two polynomials.
Exceeds the Standard (SWBAT):

Examine and evaluate peer
written expressions and justify
their conclusions.

Be able to discuss which
measure of central tendency is best
used in a given situation.
combined into one large
beach. Are there more
stars in the universe or
grains of sand on the
world’s beaches?
Explain your reasoning.
Students are given a
paused frame of the
popular video game
Tetris. Students need to
complete an entire row
to score 1 pt. for each
square in that row.
Using only the shapes
shown, students need to
determine the
maximum possible
score for the game and
explain their reasoning.
http://www.purplemath.com/modules/simpexpo.htm
http://www.math-play.com/Exponents-Jeopardy/ExponentsJeopardy.htm
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205
http://www.mathtv.com/videos_by_topic
http://www.khanacademy.org/
Typical Assessment Question(s) or Task(s):
9
Keansburg School District
Curriculum Management System
Subject/Grade/Level:
Mathematics/Algebra I
Timeline:
April-June (Year)
January (Block)
Topic(s): Equations and Inequalities
Suggested Days of
Instruction
Significance of Learning Goal(s): Solve Equations and Inequalities given any situation
Content
Standards /
CPI / Essential
Questions
CPI:
A.CED.1
A.CED.2
A.CED.3
A.CED.4
A.CED.4
A.REI.1
A.REI.10
A.REI.11
A.REI.2
A.REI.3
A.REI.4
EQ:
When does
solving
equations enter
a real-life
situation?
How are
solving
equations and
inequalities
similar?
Specific Learning Objective(s)
Suggested Activities
Instructional Tools / Materials / Technology / Resources /
Assessments and Assessment Models
The Students Will Be Able To:
Concept(s):
II.
Equations and
Inequalities
o
Solve one step
equations
o
Solve multi-step
equations
o
Solve equations with
absolute value
o
Solve and graph
simple inequalities
o
Solve and graph
compound inequalities
o
Solve and graph
inequalities with absolute value
o
Solve and graph
inequalities in two variables
Meets the Standard (SWBAT):

Solve equations involving
addition, subtraction,
multiplication or division

Solve equations using
more than one step and with
grouping symbols

Solve equations with
absolute value
Meets Standard:

Solve equations as a
group – one student at a time
with the last student checking
the result.

Using a graphing
calculator analyze graphs and
their matching inequalities.
Match graph with inequality.

Write a real-world
problem that you can model with
the two-step equation 8b + 6 =
38. Then solve the problem

Analyze the equation 24=5(g + 3). Student A decides
to divide by 5 and student B
distributes 5 first. Discuss the
two methods and determine a
preference
Graphing calculator
http://www.gamequarium.com/equations.html
http://www.khanacademy.org/math/algebra/solving-linearequations/v/equations-3
http://hotmath.com/learning_activities/algebra1a_activities.html
http://www.mathtv.com/videos_by_topic
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205
http://www.khanacademy.org/
Exceeds Standard:

Set up a spreadsheet to
solve 7(x + 1) = 3(x -1)
Does your spreadsheet show the
10

Solve and graph
inequalities involving addition,
subtraction, multiplication or
division

Solve and graph
inequalities with “and” or “or”

Solve and graph
inequalities involving absolute
value

Solve and graph
inequalities with two variables
Exceeds the Standard (SWBAT):

Solve multi-step
equations with rational numbers
with different denominators.
solution of your equation and
between which two values of x is
the solution of the equation.
Explain your reasoning. For
what spreadsheet values of x is
7(x +1) less than 3(x-1)?
* Decide whether a given
inequality is true for all real
numbers if the inequality is not
true, give a counter example.
Typical Assessment Question(s) or Task(s):
11
Keansburg School District
Curriculum Management System
Subject/Grade/Level:
Mathematics/Algebra I
Timeline:
February-April (Year)
November-December (Block)
Topic(s): Graphing Linear Equations
Suggested Days of
Instruction
Significance of Learning Goal(s):
To find slope and intercepts and write equations in different forms as pertaining real world situations.
Content
Standards /
CPI / Essential
Questions
CPI:
A.CED.2
A.REI.10
A.REI.11
A.REI.12
F.IF.7
F.IF.7A
F.IF.9
F.LE.1A
F.LE.1B
S.ID.7
S.ID.9
EQ:
What
information
does the
equation of the
line give you?
When is using
the point-slope
form more
advantageous
than using the
Specific Learning Objective(s)
Suggested Activities
The Students Will Be Able To:
Concept(s):
III.
Graphing Linear
Equations
o
Find slope
o
Identify the x
and y intercepts
o
Write equations
in point-slope form
o
Write equations
in slope-intercept form
o
Graph equations
using the x and y intercepts
o
Graph equations
using the slope and y intercept
o
Determine
whether two lines are parallel or
perpendicular
Meets the Standard (SWBAT):

Find the slope of a line
given an equation or graph.

Find the x and y
intercepts give an equation or
graph

Write an equation in
Meets Standard:

Give students slips
of paper with point-slope
form or slope-intercept
form, the slope and yintercept and a graph. Have
them line each equation
with their slope, y-intercept
and graph.

How does finding a
line’s slope by counting
units of vertical and
horizontal change on a
graph compare with finding
it using the slope formula?
Exceeds Standard:
* In a rectangle opposites
sides are parallel and
adjacent sides are
perpendicular. Figure ABCD
has vertices A (-3,3), B (-1,2), C (4,0), D(2,5). Show
Instructional Tools / Materials / Technology / Resources /
Assessments and Assessment Models
Graphing calculator
http://www.mathwarehouse.com/algebra/linear_equation/interactiveslope.php
http://www.mathwarehouse.com/algebra/linear_equation/parallelperpendicular-lines.php
http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf
http://www.mathtv.com/videos_by_topic
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205
http://www.khanacademy.org/
12
slope-intercept
form in writing
equations?
point slope form given a point
and the slope or given two points

Write an equation in
slope intercept form given the y
intercept and the slope, given a
point and the slope or given two
points

Graph a linear equation
given by finding the x and y
intercepts, the slope and one
point or two points

Determine whether two
line are parallel or perpendicular
without graphing (using the
slopes and y-intercept)
that ABCD is a rectangle.
Typical Assessment Question(s) or Task(s):

Investigate y=mx+b
by using a graphing
calculator to compare three
different equations of lines
on the same screen. How
does the sign of m affect the
graph of the equation? How
does changing the value of b
affect the graph of an
equation in form y=mx + b.
Exceeds the Standard
(SWBAT):

Change point-slope form
to standard and slope-intercept
form.
13
Keansburg School District
Curriculum Management System
Subject/Grade/Level:
Mathematics/Algebra I
Timeline:
February-April (Year)
November-December (Block)
Topic(s): Data Analysis
Suggested Days of
Instruction
Significance of Learning Goal(s): To use different measures to interpret and compare sets of data
Content
Standards /
CPI / Essential
Questions
CPI:
S.ID.1
S.ID.2
S.ID.3
S.ID.4
S.ID.5
S.ID.6
S.ID.6A
S.ID.6B
S.ID.6C
S.ID.8
EQ:
Which measure
of central
tendency is
used to discuss
your grades?
Salaries?
Popularity of a
type of music?
Specific Learning Objective(s)
Suggested Activities
Instructional Tools / Materials / Technology /
Resources / Assessments and Assessment Models
The Students Will Be Able To:
Concept(s):
IV.
Data Analysis
o
Display and Interpret
Data
o
Find Measures of
Central Tendency
o
Graph and Interpret
Scatter Plots
o
Find and graph the
Line of Best Fit
Meets the Standard (SWBAT):

Find the mean, mode, median
and range of a set of data

Graph and Interpret a scatter
plot from a set of data

Find and graph the line of best
fit

Create and interpret
histograms

Create and interpret line
graphs

Create and interpret frequency
tables
Meets Standard:

Find measures of central
tendency from a given set of data.

Make a line graph and
histogram from a given set of data
Graphing calculator
http://www.shodor.org/interactivate/activities/PlopIt/
http://illuminations.nctm.org/LessonDetail.aspx?ID=L371
http://www.quia.com/rr/51667.html
Exceeds Standard:

Make a questionnaire and
collect data to find school wide
opinion on a topic. Give hypothesis
and conclusion, creating graphs
and giving information on how
conclusion was achieved.
* Write an essay discussing which
measure of central tendency is best
used in different situations and
why.
http://nces.ed.gov/nceskids/createagraph/default.aspx
http://www.mathtv.com/videos_by_topic
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205
http://www.khanacademy.org/
14

Create and interpret stem and
leaf plots
Exceeds the Standard (SWBAT):

Create a poll and create a
scatter plot from that data.

Interpret results from a poll
stating agreement/disagreement from
main topic.
Typical Assessment Question(s) or Task(s):
15
Keansburg School District
Curriculum Management System
Subject/Grade/Level:
Mathematics/Algebra I
Timeline:
November - January (Year)
October- November (Block)
Topic(s): Functions
Suggested Days of
Instruction
Significance of Learning Goal(s): Formalizing Relations and Functions
Content
Standards
/ CPI /
Essential
Questions
CPI:
A.CED.2
A.REI.10
A.REI.11
F.IF.1
F.IF.2
F.IF.3
F.IF.4
F.IF.5
F.IF.6
F.IF.7
F.LE.5
S.ID.6A
EQ:
What is the
difference
between
direct and
inverse
variation?
When is
one better
Specific Learning
Objective(s)
Suggested
Activities
Instructional Tools / Materials / Technology / Resources / Assessments and
Assessment Models
The Students Will Be
Able To:
Concept(s):
V.
Functions
o
Find
the domain and range
of a function
o
Use
Function Notation
o
Graph
ing functions using a
given domain
o
Deter
mine whether a
relation is a function
o
Solve
problems involving
direct and inverse
variation
Meets the Standard
(SWBAT):

Identify and
match vocabulary of all
concepts related to
graphing linear
Meets Standard:

Graph
functions from a
given domain or
range.

Use the
vertical line test
to prove a graph
is a function.

Use
function notation
to simplify
expressions.
Exceeds
Standard:

Design a
roller coaster
using Roller
Coaster Tycoon on
the computer.
Find the various
Graphing calculator
Roller Coaster Tycoon
http://www.pbs.org/teachers/connect/resources/7886/preview/
http://illuminations.nctm.org/LessonsList.aspx?grade=3&standard=all
http://www.mathtv.com/videos_by_topic
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205
http://www.khanacademy.org/
Typical Assessment Question(s) or Task(s):
16
than the
other?
How are
functions
related to
equations?
equations and function
notation

Interpret and
analyze functions
using different
representations

Graph
functions on a
coordinate plane from
a given domain

Solve a direct
or inverse variation
problem
Exceeds the Standard
(SWBAT):

Solve and
graph equations with
function notation,
using slope-intercept
form.
G-forces of each
loop using
velocity and
radius
measurements
given by the
program.

You have
3 quarts of paint
to paint a room in
your house. A
quart of paint
covers a hundred
square feet. The
function A (q) =
110q represents
the area A (q), in
square feet, that q,
quarts of paint
cover. What
domain and range
are reasonable for
the function?
What is the graph
of the function?
17
Keansburg School District
Curriculum Management System
Suggested Days of
Instruction
Subject/Grade/Level:
Mathematics/Algebra I
Content
Standards
/ CPI /
Essential
Questions
CPI:
A.SSE.1A
A.SSE.1B
A.SSE.2
F.IF.8A
EQ:
How do
you tell
when
you’ve
factored a
polynomial
completely
?
Timeline:
)
Topic(s):
Significance of Learning Goal(s):
Specific Learning
Objective(s)
The Students Will Be
Able To:
Concept(s):
VI.
Factoring
o
Find
factors of monomials
o
Find
Greatest common
factor
o
Factor
using distribution
o
Factor
trinomials
o
Factor
perfect squares
Meets the Standard
(SWBAT):

Use a
factor tree to find the
factors of a monomial

Find
the Greatest common
factor of a monomial

Find
the greatest common
factor of binomials and
trinomials

Factor
Suggested
Activities
Meets Standard:
*Give polynomials
and factored
answers. Have
students match
polynomial with
its factored
counterpart.
*Set aside a 10 x
45 part of a
rectangular 45 by
x plot of land for a
garden and seed
the rest of the plot
with grass. Grass
seed costs $.03
per square foot.
Write an
expression for the
total cost of the
seed. Suppose
you buy $50.00
worth of seed,
how wide can the
area of grass be?
Explain your
Instructional Tools / Materials / Technology / Resources / Assessments and
Assessment Models
Graphing calculator
http://www.youtube.com/watch?v=z6hCu0EPs-o
http://interactive.onlinemathlearning.com/quad_factor.php?action=generate&numProblem
s=10
http://www.mathtv.com/videos_by_topic
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205
http://www.khanacademy.org/
18
trinomials of the form
X2 + BX + C

Factor
trinomials of the form
AX2 + BX + C

Factor
perfect square
binomials

Factor
perfect square
trinomials
Exceeds the Standard
(SWBAT):

Factor
polynomials with more
than three terms.
reasoning.
Typical Assessment Question(s) or Task(s):
Exceeds
Standard:
*Give a
polynomial with 4
terms. Use
factoring to
simplify.
* Paint the outside
of a jewelry box,
including the
bottom. To find
the surface area
(SA) of the
jewelry box, you
can use the
formula S.A. = 2wl
+ 2lh + 2wh,
where l = 2x +5,
w= x and h= x+3.
What is the
surface area of the
jewelry box in
terms of x? If the
surface area is
146 square units
find x.
19
Keansburg School District
Curriculum Management System
Suggested Days of
Instruction
Subject/Grade/Level:
Mathematics/Algebra I
Content
Standards /
CPI /
Essential
Questions
CPI:
A.CED.3
A.REI.12
A.REI.5
A.REI.6
A.REI.7
EQ:
Can systems
of equations
model reallife
situations?
Timeline:
Topic(s):
Significance of Learning Goal(s):
Specific Learning Objective(s)
Suggested Activities
Instructional Tools / Materials / Technology /
Resources / Assessments and Assessment Models
The Students Will Be Able To:
Concept(s):
VII.
Systems of Equations
o
Solve systems of
equations by elimination
o
Solve systems of
equations by graphing
o
Solve systems of
equations using substitution
o
Solve systems of
inequalities
Meets the Standard (SWBAT):

Solve systems of equations by
multiplying then adding or subtracting

Solve systems of equations by
graphing and finding the point of
intersection.

Solve systems of equations by
solving for one variable then
substituting

Solve systems of inequalities by
graphing and shading
Exceeds the Standard (SWBAT):

Solve a system of one equation
and one inequality.
Meets Standard:
Give a sampling of systems of
equations graphs of systems of
equations. Have students match
system with the graph.
Have groups of students solve
systems of equations by elimination.
First team to complete gets extra
credit points on next test.
Exceeds Standard:

One equation in a system is
y = 1/2x -2.
a.
Write a second equation so
that the system has one solution.
b.
Write a second equation so
that the system has no solution.
c.
Write a second equation so
that the system has infinitely many
solutions.
http://www.math-play.com/System-of--EquationsGame.html
http://www.mathtv.com/videos_by_topic
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205
http://www.khanacademy.org/
Typical Assessment Question(s) or Task(s):
* Two students are walking along
the The Henry Hudson Trail and the
20
first student starts at point 6 miles
from the beginning of the trail at
walks at a speed of 4 mph. At the
same time the second student starts
one mile from the beginning and
walks at a speed at 3 miles per hour.
What is the system of equations that
models this situation? Graph the two
equations and find the intersection.
Is this intersection point meaningful
in this situation? Explain.
21
Keansburg School District
Curriculum Management System
Suggested Days of
Instruction
Subject/Grade/Level:
Mathematics/Algebra I
Content
Standards /
CPI / Essential
Questions
CPI:
A.REI.4
A.SSE.3C
F.IF.7A
F.IF.8
F.IF.8A
F.IF.8B
F.LE.1A
F.LE.1B
F.LE.1C
F.LE.5
N.RN.2
S.ID.6A
EQ:
In which
situations can
exponential
growth or
decay be used
in real life?
Timeline:
Topic(s):
Significance of Learning Goal(s):
Specific Learning Objective(s)
Suggested Activities
The Students Will Be Able To:
Concept(s):
VIII.
Quadratic and Exponential Functions
o
Graph quadratic equations
o
Solve quadratic equations by
graphing
o
Solve quadratic equations by
factoring
o
Solve quadratic equations by
completing the square
o
Solve quadratic equations
using the quadratic formula
o
Graph exponential functions
Meets the Standard (SWBAT):

Graph quadratic equations using the
axis of symmetry and the vertex

Graph quadratic equations using a
graphing calculator

Find the roots of a quadratic equation
using the graphing calculator

Find the roots of a quadratic equation
by factoring

Find the roots of a quadratic equation
by completing the square

Find the roots of a quadratic equation
Instructional Tools / Materials /
Technology / Resources /
Assessments and Assessment Models
Meets Standard:
Graph quadratic equations on calculator. Use
the trace button to find the roots.
Ball Bounce Lab – Would different balls
bounce at different heights and rates?
Students will make charts and discuss why
one ball bounced differently than another.
Typical Assessment Question(s) or
Task(s):
Show exponential growth by giving students
liquid (some with a chemical that changes
color in it) and have them pour a little into 3
other students cups. They will then graph the
data to show how quickly disease can be
spread.
Use a die to decide how a quadratic equation
must be solved (i.e.: Rolling a 1 means to solve
by graphing, 2 means by factoring).
Exceeds Standard:
Model and predict stock market performance.
Using function notation, students will create a
function to predict future stock market
22
using the quadratic formula

Graph exponential functions with a
given domain
Exceeds the Standard (SWBAT):

Graph a quadratic equation without
the use of a calculator, finding the maximum or
minimum value and its roots.
movement.
* Using different lengths of rubber bands and
geo boards create rectangles. Make a table
showing their dimensions and calculate area.
Graph the data. Explain why the data you
collected is not linear or an exponential
function.
23
Alignment Matrices of Common Core State Standards
Common Core State Standards Vocabulary
Addition and subtraction within 5, 10, 20, 100, or 1000. Addition or subtraction of two whole numbers with whole number answers, and
with sum or minuend in the range 0-5, 0-10, 0-20, or 0-100, respectively. Example: 8 + 2 = 10 is an addition within 10, 14 – 5 = 9 is a
subtraction within 20, and 55 – 18 = 37 is a subtraction within 100.
Additive inverses. Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and – 3/4 are additive inverses of one
another because 3/4 + (– 3/4) = (– 3/4) + 3/4 = 0.
Associative property of addition. See Table 3 in this Glossary.
Associative property of multiplication. See Table 3 in this Glossary.
Bivariate data. Pairs of linked numerical observations. Example: a list of heights and weights for each player on a football team.
Box plot. A method of visually displaying a distribution of data values by using the median, quartiles, and extremes of the data set. A box
shows the middle 50% of the data.1
Commutative property. See Table 3 in this Glossary.
Complex fraction. A fraction A/B where A and/or B are fractions (B nonzero).
Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps
are carried out correctly. See also: computation strategy.
Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at
converting one problem into another. See also: computation algorithm.
Congruent. Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations,
reflections, and translations).
Counting on. A strategy for finding the number of objects in a group without having to count every member of the group. For example, if a
stack of books is known to have 8 books and 3 more books are added to the top, it is not necessary to count the stack all over again. One can
find the total by counting on—pointing to the top book and saying “eight,” following this with “nine, ten, eleven. There are eleven books now.”
24
Dot plot. See: line plot.
Dilation. A transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distances from
the center by a common scale factor.
Expanded form. A multi-digit number is expressed in expanded form when it is written as a sum of single-digit multiples of powers of ten. For
example, 643 = 600 + 40 + 3.
Expected value. For a random variable, the weighted average of its possible values, with weights given by their respective probabilities.
First quartile. For a data set with median M, the first quartile is the median of the data values less than M. Example: For the data set {1, 3, 6, 7,
10, 12, 14, 15, 22, 120}, the first quartile is 6.2 See also: median, third quartile, interquartile range.
Fraction. A number expressible in the form a/b where a is a whole number and b is a positive whole number. (The word fraction in these
standards always refers to a non-negative number.) See also: rational number.
Identity property of 0. See Table 3 in this Glossary.
Independently combined probability models. Two probability models are said to be combined independently if the probability of each
ordered pair in the combined model equals the product of the original probabilities of the two individual outcomes in the ordered pair.
Integer. A number expressible in the form a or –a for some whole number a.
Interquartile Range. A measure of variation in a set of numerical data, the interquartile range is the distance between the first and third
quartiles of the data set. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the interquartile range is 15 – 6 = 9. See also: first quartile,
third quartile.
Line plot. A method of visually displaying a distribution of data values where each data value is shown as a dot or mark above a number line.
Also known as a dot plot.3
Mean. A measure of center in a set of numerical data, computed by adding the values in a list and then dividing by the number of values in the
list.4 Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is 21.
Mean absolute deviation. A measure of variation in a set of numerical data, computed by adding the distances between each data value and
the mean, then dividing by the number of data values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean absolute deviation
is 20.
25
Median. A measure of center in a set of numerical data. The median of a list of values is the value appearing at the center of a sorted version of
the list—or the mean of the two central values, if the list contains an even number of values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15,
22, 90}, the median is 11.
Midline. In the graph of a trigonometric function, the horizontal line halfway between its maximum and minimum values.
Multiplication and division within 100. Multiplication or division of two whole numbers with whole number answers, and with product or
dividend in the range 0-100. Example: 72 ÷ 8 = 9.
Multiplicative inverses. Two numbers whose product is 1 are multiplicative inverses of one another. Example: 3/4 and 4/3 are multiplicative
inverses of one another because 3/4 × 4/3 = 4/3 × 3/4 = 1.
Number line diagram. A diagram of the number line used to represent numbers and support reasoning about them. In a number line diagram
for measurement quantities, the interval from 0 to 1 on the diagram represents the unit of measure for the quantity.
Percent rate of change. A rate of change expressed as a percent. Example: if a population grows from 50 to 55 in a year, it grows by 5/50 =
10% per year.
Probability distribution. The set of possible values of a random variable with a probability assigned to each.
Properties of operations. See Table 3 in this Glossary.
Properties of equality. See Table 4 in this Glossary.
Properties of inequality. See Table 5 in this Glossary.
Properties of operations. See Table 3 in this Glossary.
Probability. A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (such as tossing a coin,
selecting a person at random from a group of people, tossing a ball at a target, or testing for a medical condition).
Probability model. A probability model is used to assign probabilities to outcomes of a chance process by examining the nature of the
process. The set of all outcomes is called the sample space, and their probabilities sum to 1. See also: uniform probability model.
Random variable. An assignment of a numerical value to each outcome in a sample space.
Rational expression. A quotient of two polynomials with a non-zero denominator.
26
Rational number. A number expressible in the form a/b or – a/b for some fraction a/b. The rational numbers include the integers.
Rectilinear figure. A polygon all angles of which are right angles.
Rigid motion. A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid
motions are here assumed to preserve distances and angle measures.
Repeating decimal. The decimal form of a rational number. See also: terminating decimal.
Sample space. In a probability model for a random process, a list of the individual outcomes that are to be considered.
Scatter plot. A graph in the coordinate plane representing a set of bivariate data. For example, the heights and weights of a group of people
could be displayed on a scatter plot.5
Similarity transformation. A rigid motion followed by a dilation.
Tape diagram. A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model,
fraction strip, or length model.
Terminating decimal. A decimal is called terminating if its repeating digit is 0.
Third quartile. For a data set with median M, the third quartile is the median of the data values greater than M. Example: For the data set {2, 3,
6, 7, 10, 12, 14, 15, 22, 120}, the third quartile is 15. See also: median, first quartile, interquartile range.
Transitivity principle for indirect measurement. If the length of object A is greater than the length of object B, and the length of object B is
greater than the length of object C, then the length of object A is greater than the length of object C. This principle applies to measurement of
other quantities as well.
Uniform probability model. A probability model which assigns equal probability to all outcomes. See also: probability model.
Vector. A quantity with magnitude and direction in the plane or in space, defined by an ordered pair or triple of real numbers.
Visual fraction model. A tape diagram, number line diagram, or area model.
Whole numbers. The numbers 0, 1, 2, 3, ….5
27
9-12
N
RN.1
9-12
N
RN.2
9-12
N
RN.3
9-12
N
Q.1
9-12
N
Q.2
9-12
N
Q.3
9-12
N
CN.1
9-12
N
CN.2
9-12
N
CN.3
9-12
N
CN.4
Standard
CC.9-12.N.RN.1 Extend the properties of exponents to rational exponents. 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 5^(1/3) to be the cube root of 5 because we want
[5^(1/3)]^3 = 5^[(1/3) x 3] to hold, so [5^(1/3)]^3 must equal 5.
CC.9-12.N.RN.2 Extend the properties of exponents to rational exponents. Rewrite
expressions involving radicals and rational exponents using the properties of exponents.
CC.9-12.N.RN.3 Use properties of rational and irrational numbers. Explain why the sum or
product of 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.
CC.9-12.N.Q.1 Reason quantitatively and use units to solve problems. Use units as a way to
understand problems and to guide the solution of multi-step problems; choose and
interpret units consistently in formulas; choose and interpret the scale and the origin in
graphs and data displays.*
CC.9-12.N.Q.2 Reason quantitatively and use units to solve problems. Define appropriate
quantities for the purpose of descriptive modeling.*
CC.9-12.N.Q.3 Reason quantitatively and use units to solve problems. Choose a level of
accuracy appropriate to limitations on measurement when reporting quantities.*
CC.9-12.N.CN.1 Perform arithmetic operations with complex numbers. 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.
CC.9-12.N.CN.2 Perform arithmetic operations with complex numbers. Use the relation i2
= –1 and the commutative, associative, and distributive properties to add, subtract, and
multiply complex numbers.
CC.9-12.N.CN.3 (+) Perform arithmetic operations with complex numbers. Find the
conjugate of a complex number; use conjugates to find moduli and quotients of complex
numbers.
CC.9-12.N.CN.4 (+) Represent complex numbers and their operations on the complex
plane. Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar forms
of a given complex number represent the same number.
Other
Standard #
Algebra II
Strand
Geometry
Grade
Algebra I
Common Core Standards for Mathematics
Common Core State Standards for Mathematics (Grades 9-12)
X
X
X
X
X
X
X
x
X
28
9-12
N
CN.5
9-12
N
CN.6
9-12
N
CN.7
9-12
N
CN.8
9-12
N
CN.9
9-12
N
VM.1
9-12
N
VM.2
9-12
N
VM.3
9-12
N
VM.4
9-12
N
VM.4a
9-12
N
VM.4b
9-12
N
VM.4c
9-12
N
VM.5
9-12
N
VM.5a
9-12
N
VM.5b
CC.9-12.N.CN.5 (+) Represent complex numbers and their operations on the complex
plane. Represent addition, subtraction, multiplication, and conjugation of complex
numbers geometrically on the complex plane; use properties of this representation for
computation. For example, (-1 + √3i)^3 = 8 because (-1 + √3i) has modulus 2 and
argument 120°.
CC.9-12.N.CN.6 (+) Represent complex numbers and their operations on the complex
plane. Calculate the distance between numbers in the complex plane as the modulus of the
difference, and the midpoint of a segment as the average of the numbers at its endpoints.
CC.9-12.N.CN.7 Use complex numbers in polynomial identities and equations. Solve
quadratic equations with real coefficients that have complex solutions.
CC.9-12.N.CN.8 (+) Use complex numbers in polynomial identities and equations. Extend
polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x –
2i).
CC.9-12.N.CN.9 (+) Use complex numbers in polynomial identities and equations. Know
the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
CC.9-12.N.VM.1 (+) Represent and model with vector quantities. Recognize vector
quantities as having both magnitude and direction. Represent vector quantities by
directed line segments, and use appropriate symbols for vectors and their magnitudes
(e.g., v(bold), |v|, ||v||, v(not bold)).
CC.9-12.N.VM.2 (+) Represent and model with vector quantities. Find the components of a
vector by subtracting the coordinates of an initial point from the coordinates of a terminal
point.
CC.9-12.N.VM.3 (+) Represent and model with vector quantities. Solve problems involving
velocity and other quantities that can be represented by vectors.
CC.9-12.N.VM.4 (+) Perform operations on vectors. Add and subtract vectors.
CC.9-12.N.VM.4a (+) Add vectors end-to-end, component-wise, and by the parallelogram
rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the
magnitudes.
CC.9-12.N.VM.4b (+) Given two vectors in magnitude and direction form, determine the
magnitude and direction of their sum.
CC.9-12.N.VM.4c (+) Understand vector subtraction v – w as v + (–w), where (–w) is the
additive inverse of w, with the same magnitude as w and pointing in the opposite
direction. Represent vector subtraction graphically by connecting the tips in the
appropriate order, and perform vector subtraction component-wise.
CC.9-12.N.VM.5 (+) Perform operations on vectors. Multiply a vector by a scalar.
CC.9-12.N.VM.5a (+) Represent scalar multiplication graphically by scaling vectors and
possibly reversing their direction; perform scalar multiplication component-wise, e.g., as
c(v(sub x), v(sub y)) = (cv(sub x), cv(sub y)).
CC.9-12.N.VM.5b (+) Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v.
Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along
X
X
29
9-12
N
VM.6
9-12
N
VM.7
9-12
N
VM.8
9-12
N
VM.9
9-12
N
VM.10
9-12
N
VM.11
9-12
N
VM.12
9-12
A
SSE.1
9-12
A
SSE.1a
9-12
A
SSE.1b
9-12
A
SSE.2
9-12
A
SSE.3
9-12
A
SSE.3a
9-12
A
SSE.3b
9-12
A
SSE.3c
v (for c > 0) or against v (for c < 0).
CC.9-12.N.VM.6 (+) Perform operations on matrices and use matrices in applications. Use
matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network.
CC.9-12.N.VM.7 (+) Perform operations on matrices and use matrices in applications.
Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a
game are doubled.
CC.9-12.N.VM.8 (+) Perform operations on matrices and use matrices in applications. Add,
subtract, and multiply matrices of appropriate dimensions.
CC.9-12.N.VM.9 (+) Perform operations on matrices and use matrices in applications.
Understand that, unlike multiplication of numbers, matrix multiplication for square
matrices is not a commutative operation, but still satisfies the associative and distributive
properties.
CC.9-12.N.VM.10 (+) Perform operations on matrices and use matrices in applications.
Understand that the zero and identity matrices play a role in matrix addition and
multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a
square matrix is nonzero if and only if the matrix has a multiplicative inverse.
CC.9-12.N.VM.11 (+) Perform operations on matrices and use matrices in applications.
Multiply a vector (regarded as a matrix with one column) by a matrix of suitable
dimensions to produce another vector. Work with matrices as transformations of vectors.
CC.9-12.N.VM.12 (+) Perform operations on matrices and use matrices in applications.
Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value
of the determinant in terms of area.
CC.9-12.A.SSE.1 Interpret the structure of expressions. Interpret expressions that
represent a quantity in terms of its context.*
CC.9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.*
CC.9-12.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.*
CC.9-12.A.SSE.2 Interpret the structure of expressions. Use the structure of an expression
to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus
recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).
CC.9-12.A.SSE.3 Write expressions in equivalent forms to solve problems. Choose and
produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.*
CC.9-12.A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it
defines.*
CC.9-12.A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.*
CC.9-12.A.SSE.3c Use the properties of exponents to transform expressions for exponential
X
X
X
X
X
X
X
X
x
30
9-12
A
SSE.4
9-12
A
APR.1
9-12
A
APR.2
9-12
A
APR.3
9-12
A
APR.4
9-12
A
APR.5
9-12
A
APR.6
9-12
A
APR.7
9-12
A
CED.1
9-12
A
CED.2
9-12
A
CED.3
functions. For example the expression 1.15^t can be rewritten as [1.15^(1/12)]^(12t) ≈
1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate
is 15%.*
CC.9-12.A.SSE.4 Write expressions in equivalent forms to solve problems. Derive the
formula for the sum of a finite geometric series (when the common ratio is not 1), and use
the formula to solve problems. For example, calculate mortgage payments.*
CC.9-12.A.APR.1 Perform arithmetic operations on polynomials. Understand that
polynomials form a system analogous to the integers, namely, they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply
polynomials.
CC.9-12.A.APR.2 Understand the relationship between zeros and factors of polynomial.
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
CC.9-12.A.APR.3 Understand the relationship between zeros and factors of polynomials.
Identify zeros of polynomials when suitable factorizations are available, and use the zeros
to construct a rough graph of the function defined by the polynomial.
CC.9-12.A.APR.4 Use polynomial identities to solve problems. Prove polynomial identities
and use them to describe numerical relationships. For example, the polynomial identity
(x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
CC.9-12.A.APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n, where x and y are any numbers, with coefficients
determined for example by Pascal’s Triangle.1
CC.9-12.A.APR.6 Rewrite rational expressions. Rewrite simple rational expressions in
different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and
r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection,
long division, or, for the more complicated examples, a computer algebra system.
CC.9-12.A.APR.7 (+) Rewrite rational expressions. Understand that rational expressions
form a system analogous to the rational numbers, closed under addition, subtraction,
multiplication, and division by a nonzero rational expression; add, subtract, multiply, and
divide rational expressions.
CC.9-12.A.CED.1 Create equations that describe numbers or relationship. 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.*
CC.9-12.A.CED.2 Create equations that describe numbers or relationship. Create equations
in two or more variables to represent relationships between quantities; graph equations
on coordinate axes with labels and scales.*
CC.9-12.A.CED.3 Create equations that describe numbers or relationship. Represent
constraints by equations or inequalities, and by systems of equations and/or inequalities,
and interpret solutions as viable or non-viable options in a modeling context. For example,
represent inequalities describing nutritional and cost constraints on combinations of
different foods.*
X
X
X
X
X
X
X
X
X
X
X
X
31
9-12
A
CED.4
9-12
A
REI.1
9-12
A
REI.2
9-12
A
REI.3
9-12
A
REI.4
9-12
A
REI.4a
9-12
A
REI.4b
9-12
A
REI.5
9-12
A
REI.6
9-12
A
REI.7
9-12
A
REI.8
9-12
A
REI.9
9-12
A
REI.10
9-12
A
REI.11
CC.9-12.A.CED.4 Create equations that describe numbers or relationship. 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.*
CC.9-12.A.REI.1 Understand solving equations as a process of reasoning and explain the
reasoning. 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.
CC.9-12.A.REI.2 Understand solving equations as a process of reasoning and explain the
reasoning. Solve simple rational and radical equations in one variable, and give examples
showing how extraneous solutions may arise.
CC.9-12.A.REI.3 Solve equations and inequalities in one variable. Solve linear equations
and inequalities in one variable, including equations with coefficients represented by
letters.
CC.9-12.A.REI.4 Solve equations and inequalities in one variable. Solve quadratic equations
in one variable.
CC.9-12.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.
CC.9-12.A.REI.4b Solve quadratic equations by inspection (e.g., for x^2 = 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.
CC.9-12.A.REI.5 Solve systems of equations. Prove that, given a system of two equations in
two variables, replacing one equation by the sum of that equation and a multiple of the
other produces a system with the same solutions.
CC.9-12.A.REI.6 Solve systems of equations. Solve systems of linear equations exactly and
approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
CC.9-12.A.REI.7 Solve systems of equations. 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 x^2 + y^2 =
3.
CC.9-12.A.REI.8 (+) Solve systems of equations. Represent a system of linear equations as a
single matrix equation in a vector variable.
CC.9-12.A.REI.9 (+) Solve systems of equations. Find the inverse of a matrix if it exists and
use it to solve systems of linear equations (using technology for matrices of dimension 3 ×
3 or greater).
CC.9-12.A.REI.10 Represent and solve equations and inequalities graphically. Understand
that the graph of an equation in two variables is the set of all its solutions plotted in the
coordinate plane, often forming a curve (which could be a line).
CC.9-12.A.REI.11 Represent and solve equations and inequalities graphically. Explain why
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
32
9-12
A
REI.12
9-12
F
IF.1
9-12
F
IF.2
9-12
F
IF.3
9-12
F
IF.4
9-12
F
IF.5
9-12
F
IF.6
9-12
F
IF.7
9-12
F
IF.7a
9-12
F
IF.7b
the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g.,
using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational,
absolute value, exponential, and logarithmic functions.*
CC.9-12.A.REI.12 Represent and solve equations and inequalities graphically. Graph the
solutions to a linear inequality in two variables as a half-plane (excluding the boundary in
the case of a strict inequality), and graph the solution set to a system of linear inequalities
in two variables as the intersection of the corresponding half-planes.
CC.9-12.F.IF.1 Understand the concept of a function and use function notation. Understand
that a function from one set (called the domain) to another set (called the range) assigns to
each element of the domain exactly one element of the range. If f is a function and x is an
element of its domain, then f(x) denotes the output of f corresponding to the input x. The
graph of f is the graph of the equation y = f(x).
CC.9-12.F.IF.2 Understand the concept of a function and use function notation. Use
function notation, evaluate functions for inputs in their domains, and interpret statements
that use function notation in terms of a context.
CC.9-12.F.IF.3 Understand the concept of a function and use function notation. Recognize
that sequences are functions, sometimes defined recursively, whose domain is a subset of
the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1,
f(n+1) = f(n) + f(n-1) for n ≥ 1 (n is greater than or equal to 1).
CC.9-12.F.IF.4 Interpret functions that arise in applications in terms of the context. 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.*
CC.9-12.F.IF.5 Interpret functions that arise in applications in terms of the context. 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 of personhours it takes to assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.*
CC.9-12.F.IF.6 Interpret functions that arise in applications in terms of the context.
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.*
CC.9-12.F.IF.7 Analyze functions using different representations. Graph functions
expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.*
CC.9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and
minima.*
CC.9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including
X
X
x
X
X
X
X
X
x
X
X
33
9-12
F
IF.7c
9-12
F
IF.7d
9-12
F
IF.7e
9-12
F
IF.8
9-12
F
IF.8a
9-12
F
IF.8b
9-12
F
IF.9
9-12
F
BF.1
9-12
F
BF.1a
9-12
F
BF.1b
9-12
F
BF.1c
9-12
F
BF.2
9-12
F
BF.3
step functions and absolute value functions.*
CC.9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.*
CC.9-12.F.IF.7d (+) Graph rational functions, identifying zeros and asymptotes when
suitable factorizations are available, and showing end behavior.*
CC.9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and amplitude.*
CC.9-12.F.IF.8 Analyze functions using different representations. Write a function defined
by an expression in different but equivalent forms to reveal and explain different
properties of the function.
CC.9-12.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.
CC.9-12.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 and decay.
CC.9-12.F.IF.9 Analyze functions using different representations. 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.
CC.9-12.F.BF.1 Build a function that models a relationship between two quantities. Write a
function that describes a relationship between two quantities.*
CC.9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for
calculation from a context.
CC.9-12.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.
CC.9-12.F.BF.1c (+) Compose functions. For example, if T(y) is the temperature in the
atmosphere as a function of height, and h(t) is the height of a weather balloon as a function
of time, then T(h(t)) is the temperature at the location of the weather balloon as a function
of time.
CC.9-12.F.BF.2 Build a function that models a relationship between two quantities. Write
arithmetic and geometric sequences both recursively and with an explicit formula, use
them to model situations, and translate between the two forms.*
CC.9-12.F.BF.3 Build new functions from existing functions. 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.
x
X
X
X
X
X
X
34
9-12
F
BF.4
9-12
F
BF.4a
9-12
F
BF.4b
9-12
F
BF.4c
9-12
F
BF.4d
9-12
F
BF.5
9-12
F
LE.1
9-12
F
LE.1a
9-12
F
LE.1b
9-12
F
LE.1c
9-12
F
LE.2
9-12
F
LE.3
9-12
F
LE.4
9-12
F
LE.5
9-12
F
TF.1
9-12
F
TF.2
CC.9-12.F.BF.4 Build new functions from existing functions. Find inverse functions.
CC.9-12.F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an
inverse and write an expression for the inverse. For example, f(x) =2(x^3) or f(x) =
(x+1)/(x-1) for x ≠ 1 (x not equal to 1).
CC.9-12.F.BF.4b (+) Verify by composition that one function is the inverse of another.
CC.9-12.F.BF.4c (+) Read values of an inverse function from a graph or a table, given that
the function has an inverse.
CC.9-12.F.BF.4d (+) Produce an invertible function from a non-invertible function by
restricting the domain.
CC.9-12.F.BF.5 (+) Build new functions from existing functions. Understand the inverse
relationship between exponents and logarithms and use this relationship to solve
problems involving logarithms and exponents.
CC.9-12.F.LE.1 Construct and compare linear, quadratic, and exponential models and solve
problems. Distinguish between situations that can be modeled with linear functions and
with exponential functions.*
CC.9-12.F.LE.1a Prove that linear functions grow by equal differences over equal intervals
and that exponential functions grow by equal factors over equal intervals.*
CC.9-12.F.LE.1b. Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.*
CC.9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant
percent rate per unit interval relative to another.*
CC.9-12.F.LE.2 Construct and compare linear, quadratic, and exponential models and solve
problems. Construct linear and exponential functions, including arithmetic and geometric
sequences, given a graph, a description of a relationship, or two input-output pairs
(include reading these from a table).*
CC.9-12.F.LE.3 Construct and compare linear, quadratic, and exponential models and solve
problems. Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a
polynomial function.*
CC.9-12.F.LE.4 Construct and compare linear, quadratic, and exponential models and solve
problems. For exponential models, express as a logarithm the solution to ab^(ct) = d
where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using
technology.*
CC.9-12.F.LE.5 Interpret expressions for functions in terms of the situation they
model. Interpret the parameters in a linear or exponential function in terms of a context.*
CC.9-12.F.TF.1 Extend the domain of trigonometric functions using the unit circle.
Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.
CC.9-12.F.TF.2 Extend the domain of trigonometric functions using the unit circle. Explain
how the unit circle in the coordinate plane enables the extension of trigonometric
X
X
X
X
X
X
X
X
35
9-12
F
TF.3
9-12
F
TF.4
9-12
F
TF.5
9-12
F
TF.6
9-12
F
TF.7
9-12
F
TF.8
9-12
F
TF.9
9-12
G
CO.1
9-12
G
CO.2
9-12
G
CO.3
9-12
G
CO.4
9-12
G
CO.5
functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
CC.9-12.F.TF.3 (+) Extend the domain of trigonometric functions using the unit circle. Use
special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4
and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π - x,
π + x, and 2π - x in terms of their values for x, where x is any real number.
CC.9-12.F.TF.4 (+) Extend the domain of trigonometric functions using the unit circle. Use
the unit circle to explain symmetry (odd and even) and periodicity of trigonometric
functions.
CC.9-12.F.TF.5 Model periodic phenomena with trigonometric functions. Choose
trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.*
CC.9-12.F.TF.6 (+) Model periodic phenomena with trigonometric functions. Understand
that restricting a trigonometric function to a domain on which it is always increasing or
always decreasing allows its inverse to be constructed.
CC.9-12.F.TF.7 (+) Model periodic phenomena with trigonometric functions. Use inverse
functions to solve trigonometric equations that arise in modeling contexts; evaluate the
solutions using technology, and interpret them in terms of the context.*
CC.9-12.F.TF.8 Prove and apply trigonometric identities. Prove the Pythagorean identity
(sin A)^2 + (cos A)^2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A,
and the quadrant of the angle.
CC.9-12.F.TF.9 (+) Prove and apply trigonometric identities. Prove the addition and
subtraction formulas for sine, cosine, and tangent and use them to solve problems.
CC.9-12.G.CO.1 Experiment with transformations in the plane. Know precise definitions of
angle, circle, perpendicular line, parallel line, and line segment, based on the undefined
notions of point, line, distance along a line, and distance around a circular arc.
CC.9-12.G.CO.2 Experiment with transformations in the plane. Represent transformations
in the plane using, e.g., transparencies and geometry software; describe transformations
as functions that take points in the plane as inputs and give other points as outputs.
Compare transformations that preserve distance and angle to those that do not (e.g.,
translation versus horizontal stretch).
CC.9-12.G.CO.3 Experiment with transformations in the plane. Given a rectangle,
parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that
carry it onto itself.
CC.9-12.G.CO.4 Experiment with transformations in the plane. Develop definitions of
rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments.
CC.9-12.G.CO.5 Experiment with transformations in the plane. Given a geometric figure
and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph
paper, tracing paper, or geometry software. Specify a sequence of transformations that
will carry a given figure onto another.
x
X
X
X
X
36
9-12
G
CO.6
9-12
G
CO.7
9-12
G
CO.8
9-12
G
CO.9
9-12
G
CO.10
9-12
G
CO.11
9-12
G
CO.12
9-12
G
CO.13
9-12
G
SRT.1
9-12
G
SRT.2
CC.9-12.G.CO.6 Understand congruence in terms of rigid motions. Use geometric
descriptions of rigid motions to transform figures and to predict the effect of a given rigid
motion on a given figure; given two figures, use the definition of congruence in terms of
rigid motions to decide if they are congruent.
CC.9-12.G.CO.7 Understand congruence in terms of rigid motions. Use the definition of
congruence in terms of rigid motions to show that two triangles are congruent if and only
if corresponding pairs of sides and corresponding pairs of angles are congruent.
CC.9-12.G.CO.8 Understand congruence in terms of rigid motions. Explain how the criteria
for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in
terms of rigid motions.
CC.9-12.G.CO.9 Prove geometric theorems. Prove theorems about lines and angles.
Theorems include: vertical angles are congruent; when a transversal crosses parallel lines,
alternate interior angles are congruent and corresponding angles are congruent; points on
a perpendicular bisector of a line segment are exactly those equidistant from the
segment’s endpoints.
CC.9-12.G.CO.10 Prove geometric theorems. Prove theorems about triangles. Theorems
include: measures of interior angles of a triangle sum to 180 degrees; base angles of
isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle
is parallel to the third side and half the length; the medians of a triangle meet at a point.
CC.9-12.G.CO.11 Prove geometric theorems. Prove theorems about parallelograms.
Theorems include: opposite sides are congruent, opposite angles are congruent, the
diagonals of a parallelogram bisect each other, and conversely, rectangles are
parallelograms with congruent diagonals.
CC.9-12.G.CO.12 Make geometric constructions. Make formal geometric constructions with
a variety of tools and methods (compass and straightedge, string, reflective devices, paper
folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting
a segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a given line
through a point not on the line.
CC.9-12.G.CO.13 Make geometric constructions. Construct an equilateral triangle, a
square, and a regular hexagon inscribed in a circle.
CC.9-12.G.SRT.1 Understand similarity in terms of similarity transformations. Verify
experimentally the properties of dilations given by a center and a scale factor:
-- a. 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.
-- b. The dilation of a line segment is longer or shorter in the ratio given by the scale
factor.
CC.9-12.G.SRT.2 Understand similarity in terms of similarity transformations. 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
X
X
X
x
X
X
X
X
X
37
9-12
G
SRT.3
9-12
G
SRT.4
9-12
G
SRT.5
9-12
G
SRT.6
9-12
G
SRT.7
9-12
G
SRT.8
9-12
G
SRT.9
9-12
G
SRT.10
9-12
G
SRT.11
9-12
G
C.1
9-12
G
C.2
9-12
G
C.3
9-12
G
C.4
9-12
G
C.5
corresponding pairs of sides.
CC.9-12.G.SRT.3 Understand similarity in terms of similarity transformations. Use the
properties of similarity transformations to establish the AA criterion for two triangles to
be similar.
CC.9-12.G.SRT.4 Prove theorems involving similarity. 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.
CC.9-12.G.SRT.5 Prove theorems involving similarity. Use congruence and similarity
criteria for triangles to solve problems and to prove relationships in geometric figures.
CC.9-12.G.SRT.6 Define trigonometric ratios and solve problems involving right triangles.
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.
CC.9-12.G.SRT.7 Define trigonometric ratios and solve problems involving right triangles.
Explain and use the relationship between the sine and cosine of complementary angles.
CC.9-12.G.SRT.8 Define trigonometric ratios and solve problems involving right triangles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
CC.9-12.G.SRT.9 (+) Apply trigonometry to general triangles. Derive the formula A =
(1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side.
CC.9-12.G.SRT.10 (+) Apply trigonometry to general triangles. Prove the Laws of Sines and
Cosines and use them to solve problems.
CC.9-12.G.SRT.11 (+) Apply trigonometry to general triangles. Understand and apply the
Law of Sines and the Law of Cosines to find unknown measurements in right and non-right
triangles (e.g., surveying problems, resultant forces).
CC.9-12.G.C.1 Understand and apply theorems about circles. Prove that all circles are
similar.
CC.9-12.G.C.2 Understand and apply theorems about circles. Identify and describe
relationships among inscribed angles, radii, and chords. Include the relationship between
central, inscribed, and circumscribed angles; inscribed angles on a diameter are right
angles; the radius of a circle is perpendicular to the tangent where the radius intersects
the circle.
CC.9-12.G.C.3 Understand and apply theorems about circles. Construct the inscribed and
circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
CC.9-12.G.C.4 (+) Understand and apply theorems about circles. Construct a tangent line
from a point outside a given circle to the circle.
CC.9-12.G.C.5 Find arc lengths and areas of sectors of circles. Derive using similarity the
fact that the length of the arc intercepted by an angle is proportional to the radius, and
define the radian measure of the angle as the constant of proportionality; derive the
X
X
X
X
X
X
X
X
X
X
X
X
X
X
38
9-12
G
GPE.1
9-12
G
GPE.2
9-12
G
GPE.3
9-12
G
GPE.4
9-12
G
GPE.5
9-12
G
GPE.6
9-12
G
GPE.7
9-12
G
GMD.1
9-12
G
GMD.2
9-12
G
GMD.3
9-12
G
GMD.4
9-12
G
MG.1
formula for the area of a sector.
CC.9-12.G.GPE.1 Translate between the geometric description and the equation for a conic
section. Derive the equation of a circle of given center and radius using the Pythagorean
Theorem; complete the square to find the center and radius of a circle given by an
equation.
CC.9-12.G.GPE.2 Translate between the geometric description and the equation for a conic
section. Derive the equation of a parabola given a focus and directrix.
CC.9-12.G.GPE.3 (+) Translate between the geometric description and the equation for a
conic section. Derive the equations of ellipses and hyperbolas given the foci, using the fact
that the sum or difference of distances from the foci is constant.
CC.9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the coordinate
plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at
the origin and containing the point (0, 2).
CC.9-12.G.GPE.5 Use coordinates to prove simple geometric theorems algebraically. Prove
the slope criteria for parallel and perpendicular lines and use them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that
passes through a given point).
CC.9-12.G.GPE.6 Use coordinates to prove simple geometric theorems algebraically. Find
the point on a directed line segment between two given points that partitions the segment
in a given ratio.
CC.9-12.G.GPE.7 Use coordinates to prove simple geometric theorems algebraically. Use
coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
using the distance formula.*
CC.9-12.G.GMD.1 Explain volume formulas and use them to solve problems. 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.
CC.9-12.G.GMD.2 (+) Explain volume formulas and use them to solve problems. Give an
informal argument using Cavalieri’s principle for the formulas for the volume of a sphere
and other solid figures.
CC.9-12.G.GMD.3 Explain volume formulas and use them to solve problems. Use volume
formulas for cylinders, pyramids, cones, and spheres to solve problems.*
CC.9-12.G.GMD.4 Visualize relationships between two-dimensional and three-dimensional
objects. Identify the shapes of two-dimensional cross-sections of three-dimensional
objects, and identify three-dimensional objects generated by rotations of two-dimensional
objects.
CC.9-12.G.MG.1 Apply geometric concepts in modeling situations. Use geometric shapes,
their measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder).*
X
X
X
X
X
X
X
X
X
X
X
X
39
9-12
G
MG.2
9-12
G
MG.3
9-12
S
ID.1
9-12
S
ID.2
9-12
S
ID.3
9-12
S
ID.4
9-12
S
ID.5
9-12
S
ID.6
9-12
S
ID.6a
9-12
9-12
S
S
ID.6b
ID.6c
9-12
S
ID.7
9-12
S
ID.8
9-12
9-12
S
S
ID.9
IC.1
CC.9-12.G.MG.2 Apply geometric concepts in modeling situations. Apply concepts of
density based on area and volume in modeling situations (e.g., persons per square mile,
BTUs per cubic foot).*
CC.9-12.G.MG.3 Apply geometric concepts in modeling situations. 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).*
CC.9-12.S.ID.1 Summarize, represent, and interpret data on a single count or measurement
variable. Represent data with plots on the real number line (dot plots, histograms, and box
plots).*
CC.9-12.S.ID.2 Summarize, represent, and interpret data on a single count or measurement
variable. Use statistics appropriate to the shape of the data distribution to compare center
(median, mean) and spread (interquartile range, standard deviation) of two or more
different data sets.*
CC.9-12.S.ID.3 Summarize, represent, and interpret data on a single count or measurement
variable. Interpret differences in shape, center, and spread in the context of the data sets,
accounting for possible effects of extreme data points (outliers).*
CC.9-12.S.ID.4 Summarize, represent, and interpret data on a single count or measurement
variable. Use the mean and standard deviation of a data set to fit it to a normal distribution
and to estimate population percentages. Recognize that there are data sets for which such
a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas
under the normal curve.*
CC.9-12.S.ID.5 Summarize, represent, and interpret data on two categorical and
quantitative variables. Summarize categorical data for two categories in two-way
frequency tables. Interpret relative frequencies in the context of the data (including joint,
marginal, and conditional relative frequencies). Recognize possible associations and
trends in the data.*
CC.9-12.S.ID.6 Summarize, represent, and interpret data on two categorical and
quantitative variables. Represent data on two quantitative variables on a scatter plot, and
describe how the variables are related.*
CC.9-12.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, quadratic, and exponential models.*
CC.9-12.S.ID.6b Informally assess the fit of a function by plotting and analyzing residuals.*
CC.9-12.S.ID.6c Fit a linear function for a scatter plot that suggests a linear association.*
CC.9-12.S.ID.7 Interpret linear models. Interpret the slope (rate of change) and the
intercept (constant term) of a linear model in the context of the data.*
CC.9-12.S.ID.8 Interpret linear models. Compute (using technology) and interpret the
correlation coefficient of a linear fit.*
CC.9-12.S.ID.9 Interpret linear models. Distinguish between correlation and causation.*
CC.9-12.S.IC.1 Understand and evaluate random processes underlying statistical
X
X
X
40
9-12
S
IC.2
9-12
S
IC.3
9-12
S
IC.4
9-12
S
IC.5
9-12
S
IC.6
9-12
S
CP.1
9-12
S
CP.2
9-12
S
CP.3
9-12
S
CP.4
9-12
S
CP.5
experiments. Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.*
CC.9-12.S.IC.2 Understand and evaluate random processes underlying statistical
experiments. Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. For example, a model says a spinning coin falls
heads up with probability 0. 5. Would a result of 5 tails in a row cause you to question the
model?*
CC.9-12.S.IC.3 Make inferences and justify conclusions from sample surveys, experiments,
and observational studies. Recognize the purposes of and differences among sample
surveys, experiments, and observational studies; explain how randomization relates to
each.*
CC.9-12.S.IC.4 Make inferences and justify conclusions from sample surveys, experiments,
and observational studies. Use data from a sample survey to estimate a population mean
or proportion; develop a margin of error through the use of simulation models for random
sampling.*
CC.9-12.S.IC.5 Make inferences and justify conclusions from sample surveys, experiments,
and observational studies. Use data from a randomized experiment to compare two
treatments; use simulations to decide if differences between parameters are significant.*
CC.9-12.S.IC.6 Make inferences and justify conclusions from sample surveys, experiments,
and observational studies. Evaluate reports based on data.*
X
CC.9-12.S.CP.2 Understand independence and conditional probability and use them to
interpret data. 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.*
CC.9-12.S.CP.3 Understand independence and conditional probability and use them to
interpret data. 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.*
CC.9-12.S.CP.4 Understand independence and conditional probability and use them to
interpret data. 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 your school will favor science given that the student is in
tenth grade. Do the same for other subjects and compare the results.*
CC.9-12.S.CP.5 Understand independence and conditional probability and use them to
interpret data. Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. For example, compare the
X
X
X
x
41
9-12
S
CP.6
9-12
S
CP.7
9-12
S
CP.8
9-12
S
CP.9
9-12
S
MD.1
9-12
S
MD.2
9-12
S
MD.3
9-12
S
MD.4
9-12
S
MD.5
9-12
S
MD.5a
9-12
S
MD.5b
chance of having lung cancer if you are a smoker with the chance of being a smoker if you
have lung cancer.*
CC.9-12.S.CP.6 Use the rules of probability to compute probabilities of compound events in
a uniform probability model. 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.*
CC.9-12.S.CP.7 Use the rules of probability to compute probabilities of compound events in
a uniform probability model. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and
B), and interpret the answer in terms of the model.*
CC.9-12.S.CP.8 (+) Use the rules of probability to compute probabilities of compound
events in a uniform probability model. Apply the general Multiplication Rule in a uniform
probability model, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)], and interpret the
answer in terms of the model.*
CC.9-12.S.CP.9 (+) Use the rules of probability to compute probabilities of compound
events in a uniform probability model. Use permutations and combinations to compute
probabilities of compound events and solve problems.*
CC.9-12.S.MD.1 (+) Calculate expected values and use them to solve problems. Define a
random variable for a quantity of interest by assigning a numerical value to each event in a
sample space; graph the corresponding probability distribution using the same graphical
displays as for data distributions.*
CC.9-12.S.MD.2 (+) Calculate expected values and use them to solve problems. Calculate
the expected value of a random variable; interpret it as the mean of the probability
distribution.*
CC.9-12.S.MD.3 (+) Calculate expected values and use them to solve problems. Develop a
probability distribution for a random variable defined for a sample space in which
theoretical probabilities can be calculated; find the expected value. For example, find the
theoretical probability distribution for the number of correct answers obtained by
guessing on all five questions of a multiple-choice test where each question has four
choices, and find the expected grade under various grading schemes.*
CC.9-12.S.MD.4 (+) Calculate expected values and use them to solve problems. Develop a
probability distribution for a random variable defined for a sample space in which
probabilities are assigned empirically; find the expected value. For example, find a current
data distribution on the number of TV sets per household in the United States, and
calculate the expected number of sets per household. How many TV sets would you expect
to find in 100 randomly selected households?*
CC.9-12.S.MD.5 (+) Use probability to evaluate outcomes of decisions. Weigh the possible
outcomes of a decision by assigning probabilities to payoff values and finding expected
values.*
CC.9-12.S.MD.5a (+) Find the expected payoff for a game of chance. For example, find the
expected winnings from a state lottery ticket or a game at a fast-food restaurant.*
CC.9-12.S.MD.5b (+) Evaluate and compare strategies on the basis of expected values. For
example, compare a high-deductible versus a low-deductible automobile insurance policy
X
X
X
X
X
X
X
X
X
X
42
9-12
S
MD.6
9-12
S
MD.7
using various, but reasonable, chances of having a minor or a major accident.*
CC.9-12.S.MD.6 (+) Use probability to evaluate outcomes of decisions. Use probabilities to
make fair decisions (e.g., drawing by lots, using a random number generator).*
CC.9-12.S.MD.7 (+) Use probability to evaluate outcomes of decisions. Analyze decisions
and strategies using probability concepts (e.g., product testing, medical testing, pulling a
hockey goalie at the end of a game).*
43