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Teacher TrainingRevised ELA and Math Standards
Math 9–12Tennessee Department of Education | 2017 Summer Teacher Training
Welcome, Teachers!
1
We are excited to welcome you to this summer’s teacher training on the revised math standards. We appreciate your dedication to the students in your classroom and your growth as an educator. As you interact with the math standards over the next two days, we hope you are able to find ways to connect this new content to your own classroom. Teachers perform outstanding work every school year, and our hope is that the knowledge you gain this week will enhance the high-quality instruction you provide Tennessee's children every day.
We are honored that the content of this training was developed by and with Tennessee educators for Tennessee educators. We believe it is important for professional development to be informed by current educators, who work every day to cultivate every student’s potential.
We’d like to thank the following educators for their contribution to the creation and review of this content:
Dr. Holly Anthony, Tennessee Technological University Michael Bradburn, Alcoa City Schools Dr. Jo Ann Cady, University of Tennessee Sherry Cockerham, Johnson City Schools Dr. Allison Clark, Arlington Community Schools Kimberly Herring, Cumberland County Schools Dr. Joseph Jones, Cheatham County Schools Dr. Emily Medlock, Lipscomb University
Part 1: The Standards
3
Module 1: Standards Review Process
Module 2: Tennessee Academic Standards
Module 3: Summary of Revisions
Part 2: Developing a Deeper Understanding
Module 4: Diving into the Standards (KUD)
Part 3: Instructional Shifts
Module 5: Revisiting the Shifts and SMP’s
Module 6: Literacy Skills for Mathematical Proficiency
Part 4: Assessment and Materials
Module 7: Connecting Standards and Assessment
Module 8: Evaluating Instructional Materials
Part 5: Putting it All Together
Module 9: Instructional Planning
Notes
Time Content
8–11:15(includes break)
Part 1: The Standards• M1: Standards Review Process• M2: TN Academic Standards• M3: Summary of Revisions
11:15–12:30 Lunch (on your own)
12:30–4(includes break)
Part 2: Developing a Deeper Understanding• M4: Diving into the Standards (KUD)
Part 3: Instructional Shifts• M5: Revisiting the Shifts and SMP’s• M6: Literacy Skills for Mathematical Proficiency
Agenda: Day 1
Goals: Day 1• Review the standards revision process.
• Highlight changes/revisions to standards.
• Use a KUD exercise to deepen our understanding of the expectations of the standards.
• Explore the Literacy Skills for Mathematical Proficiency.
5
• Discuss the instructional shifts and the Standards for Mathematical Practice (SMPs).
Agenda: Day 2
Time Content
8–11:15(includes break)
Part 4: Aligned Materials and Assessments• M7: Assessing Student Understanding
11:15–12:30 Lunch (on your own)
12:30–4(includes break)
• M8: Evaluating Instructional Materials
Part 5: Putting it All Together• M9: Instructional Planning
Goals: Day 2• Examine best practices for assessing student learning.
• Develop a process for evaluating instructional materials.
• Connect standards and assessment through instructional planning.
7
Appointment TimeMake four appointments to meet with fellow participants throughout the training to discuss the content. Record participants’ names in the form below and bookmark this page for your reference.
1 2
3 4
9
Key Ideas for Teacher Training
Strong Standards High Expectations
Instructional Shifts
Aligned Materials and Assessments
11
High Expectations
We have a continued goal to prepare students to be college and career ready.
We know that Tennessee educators are working hard and striving to get better. This summer’s teacher training is an exciting opportunity to learn about our state’s newly adopted math and ELA standards and ways to develop a deeper understanding of the standards to improve classroom instructional practices. The content of this training is aligned to the standards and is designed to address the needs of educators across our state.
Throughout this training, you will find a series of key ideas that are designed to focus our work on what is truly important. These key ideas align to the training objectives and represent the most important concepts of this course.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
Instructional Shifts
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Aligned Materials and Assessments
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
13
Strong Standards
High Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 1: The StandardsModule 1: The Standards Review Process
TAB PAGE
Standards Review Process The graphic below illustrates Tennessee’s standards review process. Here you can see the various stakeholders involved throughout the process.
• The process begins with a website for public feedback. • Tennessee educators who are experts in their content area and grade band serve
on the advisory panels. These educators review all the public feedback and the current standards, then use their content expertise and knowledge of Tennessee students to draft a revised set of standards.
• The revised standards are posted for a second feedback collection from Tennessee’s stakeholders.
• The Standards Recommendation Committee (SRC) consists of 10 members appointed by legislators. This group looks at all the feedback from the website, the current standards, and revised drafts. Recommendations are then made for additional revisions if needed.
• The SRC recommends the final draft to the State Board of Education for approval.
17
Educator Advisory Team MembersEvery part of the state was represented with multiple voices.
47Timeline of Standards Adoptions and Aligned Assessments Implementation
Subject
Math
ELA
Science
Social Studies
Standards Proposed to State Board
for Final Read
April 2016
April 2016
October 2016
July 2017
LEAs: New Standards in Classrooms
2017–18
2017–18
2018–19
2019–20
Aligned Assessment
2017–18
2017–18
2018–19
2019–20
18
Standards Revision Key Points
• within a single grade level, and• between multiple grade levels.
“Districts and schools in Tennessee will exemplify excellence and equity such that
all students are equipped with the knowledge and skills to successfully
embark upon their chosen path in life.”
What is your role in ensuring that all students are college and career ready?
19
• The instructional shifts remain the same and are still the focus of the standards.• The revised standards represent a stronger foundation that will support the progression of rigorous standards throughout the grade levels.
• The revised standards improve connections:
Strong Standards
High Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 1: The StandardsModule 2: The Tennessee Mathematics Academic Standards
TAB PAGE
Goals• Reinforce the continued expectations of the Tennessee Math Academic
Standards.
• Revisit the three instructional shifts and their continued and connected role in the revised standards.
• Review the overarching changes of the revised Tennessee Math Academic Standards.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standardsarethebricksthatshouldbemasterfullylaidthroughqualityinstructiontoensurethatallstudentsreachtheexpectationofthestandards.
Instructional Shifts
Theinstructionalshiftsareanessentialcomponentofthestandardsandprovideguidanceforhowthestandardsshouldbetaughtandimplemented.
Aligned Materials and Assessments
Educatorsplayakeyroleinensuringthatourstandards,classroominstructionalmaterials,andassessmentsarealigned.
23
Setting the StageDirections:
2. After reading and annotating the two parts, write the sentence or phrase you felt was the most important in the box below and your rationale for choosing it.
Most Important Idea:
Rationale:
Key Ideas from Discussion:
24
1. Read and annotate the General Introduction to the TN Math Standards (pages 1–2) focusing on the “Mathematically Prepared” and “Conceptual Understanding, Procedural Fluency, and Application” sections.
What Has NOT Changed
• Students prepared for college and career
• K–12 learning progressions
• Traditional and integrated pathways (for high school)
• Standards for Mathematical Practice
• Instructional shifts
What HAS Changed• Category Change
• Revised Structured
• Coding & Nomenclature
• Literacy Skills for Mathematical Proficiency
Notes:
25
What HAS Changed
Category Change
Supporting Work of the Grade
Major Work of the Grade
Major Work of the Grade
Additional Work of the Grade
Supporting Work of the
Grade
Former Current
Notes:
26
What HAS Changed
Revised Structure
Notes:
27
What HAS Changed
Revised Structure
Notes:
28
What HAS Changed
Coding and Nomenclature
Notes:
A2.N.RN.A.1
M1
A
SSE
A
1
M1.A.SSE.A.1
A2
N
RN
A
1
29
Communication in mathematics requires literacy skills in reading, vocabulary, speaking, listening, and writing. Students must be able to:
What HAS Changed
Literacy Skills for Mathematical Proficiency
Notes:
30
Module 2 Review• Reinforce the continued expectations of the Tennessee Math Academic
Standards.
• Revisit the three instructional shifts and their continued and connected role in the revised standards.
• Review the overarching changes of the revised Tennessee Math Academic Standards.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
31
Strong Standards
High Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 1: The StandardsModule 3: Summary of Revisions
33
TAB PAGE
Goals
• Review a summary of revisions to the math standards by grade band.
• Compare 2016–17 standards to 2017–18 standards.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standardsarethebricksthatshouldbemasterfullylaidthroughqualityinstructiontoensurethatallstudentsreachtheexpectationofthestandards.
Instructional Shifts
Theinstructionalshiftsareanessentialcomponentofthestandardsandprovideguidanceforhowthestandardsshouldbetaughtandimplemented.
Aligned Materials and Assessments
Educatorsplayakeyroleinensuringthatourstandards,classroominstructionalmaterials,andassessmentsarealigned.
35
“To assess student achievement accurately, teachers and administrators must know and understand the content standards that their students are to master. Again, we cannot teach or assess achievement that we have not defined.”
—S. Chappuis, Stiggins, Arter, and J. Chappuis, 2006
Why Standards?
What about this quotation sticks out to you?
Notes:
36
Revisions to the Math Standards
Specific to K–5
• Refined for clarity
• Increased fluency expectations
• Revised examples
Overarching Revisions
• Supporting and additional work of the grade is combined as supporting work of the grade
• Increased fluency expectations
Increased Fluency Expectations
Former Standard Current Standard
Kindergarten K.OA.5 Fluently add and subtract within 5.
K.OA.A.5 Fluently add and subtract within 10 using mental strategies.
FirstGrade
1.OA.6. Add and subtract within 20, demonstrating fluency for addition
and subtraction within 10.
1.OA.C.6 Fluently add and subtract within 20 using mental strategies. By
the end of Grade 1, know from memory all sums up to 10.
SecondGrade
2.OA.2 Fluently add and subtract within 20 using mental strategies.
By end of Grade 2, know from memory all sums of two one-digit
numbers.
2.OA.B.2 Fluently add and subtract within 30 using mental strategies. By
the end of Grade 2, know from memory all sums of two one-digit numbers and related subtraction
facts.
37
Revisions to the Math Standards
Specific to K–5
• Refined for clarity
• Increased fluency expectations
• Revised examples
Overarching Revisions
• Added/shifted a small number of standards to strengthen coherence across grade levels
38
Revisions to the Math Standards
Specific to K–5
• Refined for clarity
• Increased fluency expectations
• Revised examples
Overarching Revisions
• Added/shifted a small number of standards to strengthen coherence across grade levels
39
Revisions to the Math Standards
Specific to K–5
• Refined for clarity
• Increased fluency expectations
• Revised examples
Overarching Revisions• Revised language to provide clarity and continuity
• Highlighted chart for–grade level mastery expectation for addition, subtraction, multiplication and division
Former Standard 2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
Current Standard 2.NBT.A.3 Read and write numbers to 1000 using standard form, word form, and expanded form.
Former Standard 4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.
Current Standard 4.NBT.A.3 Round multi-digit whole numbers to any place (up to and including the hundred-thousand place) using understanding of place value.
40
Revisions to the Math Standards
Specific to 6–8
• Refined major work of the grade
• Revised supporting work of the grade, especially in statistics and probability
Overarching Revisions
• Slight revisions made to geometry in grade 8
• Supporting and additional work of the grade is combined as supporting work of the grade
• Revised language to provide clarity and continuity
Former Standard 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
41
Current Standard 6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center (mean, median, mode), spread (range), and overall shape.
Revisions to the Math Standards
Specific to 6–8
• Refined major work of the grade
• Revised supporting work of the grade, especially in statistics and probability
Overarching Revisions
• Revised a small number of standards to strengthen coherence by condensing, expanding, and removing standards
• Revised a small number of statistics and probability standards
Former Standard 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another. For example, Susan is putting money in her savings account by depositing a set amount each week (50). Represent her savings account balance with respect to the number of weekly deposits (s = 50w, illustrating the relationship between balance amount s and number of weeks w). Write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
Current Standard 6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another. For example, Susan is putting money in her savings account by depositing a set amount each week (50). Represent her savings account balance with respect to the number of weekly deposits (s = 50w, illustrating the relationship between balance amount s and number of weeks w).
a. Write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.
b. Analyze the relationship between the dependent and independentvariables using graphs and tables, and relate these to the equation.
42
Removed Standard 7.G.3 Describe the two-dimensional figures that result from slicing three dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Revisions to the Math Standards
Specific to 6–8
• Refined major work of the grade
• Revised supporting work of the grade, especially in statistics and probability
Overarching Revisions
• Revised a small number of standards to strengthen coherence by condensing, expanding, and removing standards
• Revised a small number of statistics and probability standards
Former Standard6.SP.5c Summarize numerical data sets in relation to their context, such as by: c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
43
Current Standard 6.SP.B.5c Summarize numerical data sets in relation to their context, such as by: c. Giving quantitative measures of center (median and/or mean) and variability (range), as well as describing any overall pattern with reference to the context in which the data were gathered.
Revisions to the Math Standards
Specific to 9–12
• Refined and revised scope and clarifications
• Revisions for Algebra II and Integrated Math III
• Restructured additional math courses to reflect college and career readiness
Overarching Revisions
• Supporting and additional work of the grade is combined as supporting work of the grade
• Removed or shifted a small number of standards to the major work of the grade to streamline vertical progression
• Revised language and examples to provide clarity and continuity
Former StandardG.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Moved StandardA2/M3.F.TF.5 to P.F.TF.A.4 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
This standard moved from Algebra II/Integrated III to Pre-Calculus.
44
• Shifted a small number of supporting work of the grade standards to additional mathematics courses
Current Standard G.SRT.C.8a Know and use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Revisions to the Math Standards
Specific to 9–12
• Refined and revised scope and clarifications
• Revisions for Algebra II and Integrated Math III
Overarching Revisions
• Restructured additional mathematics courses to reflect college and career readiness by removing three courses and adding “Applied Mathematical Concepts”
Rationale:
• High expectations
• Retention of rigorous standards
• Clearly defined and coherent pathways
• Equity and opportunity
• Aligned with student interest in postsecondary fields
• Shift to a discipline and career based pathway
Former:• Advanced Algebra and Trigonometry• Discrete Math• Finite Math• Bridge Math • Pre-Calculus• Statistics• Calculus
Current:• Applied Mathematical Concepts• Bridge Math• Pre-Calculus• Statistics• Calculus
45
• Restructured additional courses to reflect college and career readiness
Revisions to the Math Standards
New Applied Mathematical Concepts Course• For students interested in careers that use applied mathematics such as banking,
industry, or human resources
• Rich problem solving experience
• Combines standards from Senior Finite Math and Discrete Mathematics
• Designed with industry needs in mind
• Alignment with first three math courses and ACT college and career readiness
• Possible dual credit exam
Problems in Applied Mathematical ConceptsAM.G.L.A.3: Solve a variety of logic puzzles
What's the easiest way to heat a pan of water for 9 minutes when you have only a 6-minute hour-glass timer and a 21-minute hour-glass timer?
AM.D.ID.A.2: Use a variety of counting methods to organize information, determine probabilities, and solve problems.
Given a group of students: G = {Allen, Brenda, Chad, Dorothy, Eric} list and count the different ways of choosing the following officers or representatives for student congress. Assume that no one can hold more than one office.
A president, a secretary, and a treasurer, if the president must be a woman and the other two must be men.
AM.N.NQ.B.6: Solve contextual problems involving financial decision-making.
The cash price of a fitness system is $659.99. The customer paid $115 as a down payment. The remainder will be paid in 36 monthly installments of $19.16 each. Find the amount of the finance charge.
46
Revisions to the Math Standards
Standards Comparison Activity
Compare the former standards to the current standards
Directions:
1. Highlight any changes you notice between the former standards and the current standards in the column on the right.
Notes:
47
2. Use the included chart to compare the former standards with the current standards.
Notation
TN Standards through M
ay 2017 Standard Clarification
Revised TN Standards
Standard Clarification
A1.N.RN.A.3 3. Explain w
hy the sum or
product of two rational
numbers is rational; and that
they product of a nonzero rational num
ber and an irrational num
ber is irrational.
There is no additional scope or clarification inform
ation for this standard.
A1.N.Q.A.1
1. Use units as a w
ay to understand problem
s and to guide the solution of m
ulti-step problem
s; choose and interpret units consistently in form
ulas; choose and interpret the scale and the origin in graphs and data displays.
There is no additional scope or clarification inform
ation for this standard.
A1.N.Q.A.1 U
se units as a way to
understand problems and to guide
the solution of multi-step problem
s; choose and interpret units consistently in form
ulas; choose and interpret the scale and the origin in graphs and data displays.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.N.Q.A.2
2. Define appropriate
quantities for the purpose of descriptive m
odeling.
This standard will be assessed in
Algebra I by ensuring that some
modeling tasks (involving A
lgebra I content or securely held content
from grades 6-8) require the student
to create a quantity of interest in the situation being described (i.e., a
quantity of interest is not selected for the student by the task). For exam
ple, in a situation involving data, the student m
ight autonom
ously decide that a m
easure of center is a key variable in a situation, and then choose to
work w
ith the mean
A1.N.Q.A
.2 Identify, interpret, and justify appropriate quantities for the purpose of descriptive m
odeling.
Descriptive m
odeling refers to understanding and interpreting graphs; identifying extraneous inform
ation; choosing appropriate units; etc. There are no assessm
ent limits for this
standard. The entire standard is assessed in this course.
A1.N.Q.A.3
3. Choose a level of accuracy
appropriate to limitations on
measurem
ent when reporting
quantities.
There is no additional scope or clarification inform
ation for this standard.
A1.N.Q.A.3 C
hoose a level of accuracy appropriate to lim
itations on m
easurement w
hen reporting quantities.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
48
A1.A
.SSE
.A.1
1.
Inte
rpre
t exp
ress
ions
that
repr
esen
t a q
uant
ity in
term
sof
its
cont
ext.★
a.
Inte
rpre
t par
ts o
f an
expr
essi
on, s
uch
as te
rms,
fact
ors,
and
coe
ffici
ents
.
b.In
terp
ret c
ompl
icat
edex
pres
sion
s by
vie
win
g on
e or
mor
e of
thei
r par
ts a
s a
sing
l een
tity.
For
exa
mpl
e, in
terp
ret
P(1+
r) n as
the
prod
uct o
f Pan
d a
fact
or n
ot d
epen
ding
on
P.
T her
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.A
.SSE
.A.1
Inte
rpre
t exp
ress
ions
th
at re
pres
ent a
qua
ntity
in te
rms
of
its c
onte
xt.★
a.
Inte
rpre
t par
ts o
f an
expr
essi
on,
such
as
term
s, fa
ctor
s, a
ndco
effic
ient
s.
b.In
terp
ret c
ompl
icat
ed e
xpre
ssio
nsby
vie
win
g on
e or
mor
e of
thei
r par
tsas
a s
ingl
e en
tity.
For e
xam
ple,
inte
rpre
t P
(1 +
r) n a
s th
e pr
oduc
t of P
and
a
fact
or n
ot d
epen
ding
on
P.
T her
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e.
A1.A
.SSE
.A.2
2.
Use
the
stru
ctur
e of
an
expr
essi
on to
iden
tify
way
s to
rew
rite
it. F
or e
xam
ple,
see
–as
,th
us re
cogn
izin
g it
as a
diffe
renc
e of
squ
ares
that
can
be fa
ctor
ed a
s
i)Ta
sks
are
limite
d to
num
eric
alex
pres
sion
s an
d po
lyno
mia
lex
pres
sion
s in
one
var
iabl
e.
i i)Ex
ampl
es: R
ecog
nize
532 -
472
as a
diff
eren
ce o
f squ
ares
and
see
an o
ppor
tuni
ty to
rew
rite
it in
the
easi
er-to
-eva
luat
e fo
rm (5
3+47
)(53
-47
). Se
e an
opp
ortu
nity
to re
writ
e a2
+9a
+ 1
4 as
(a+7
)(a-
2).
A1.A
.SSE
.A.2
Use
the
stru
ctur
e of
an
exp
ress
ion
to id
entif
y w
ays
to
rew
rite
it.
For e
xam
ple,
reco
gniz
e 5
32 - 4
72 as
a di
ffere
nce
of s
quar
es
and
see
an o
ppor
tuni
ty to
rew
rite
it in
th
e ea
sier
to-e
valu
ate
form
(53
+ 47
)(53
– 4
7). S
ee a
n op
portu
nity
to
rew
rite
a2 +
9a +
14
as (a
+ 7
)(a
+ 2)
.
Task
s ar
e lim
ited
to n
umer
ical
ex
pres
sion
s an
d po
lyno
mia
l ex
pres
sion
s in
one
var
iabl
e.
A1.A
.SSE
.B.3
3.
Cho
ose
and
prod
uce
aneq
uiva
lent
form
of a
nex
pres
sion
to re
veal
and
expl
ain
prop
ertie
s of
the
quan
tity
repr
esen
ted
by th
eex
pres
sion
. ★
a.Fa
ctor
a q
uadr
atic
expr
essi
on to
reve
al th
e ze
ros
of th
e fu
nctio
n it
defin
es.
b .C
ompl
ete
the
squa
re in
aqu
adra
tic e
xpre
ssio
n to
reve
alth
e m
axim
um o
r min
imum
v alu
e of
the
func
tion
it de
fines
.
i)Ta
sks
have
a re
al-w
orld
con
text
.As
des
crib
ed in
the
stan
dard
, the
reis
an
inte
rpla
y be
twee
n th
em
athe
mat
ical
stru
ctur
e of
the
expr
essi
on a
nd th
e st
ruct
ure
of th
esi
tuat
ion
such
that
cho
osin
g an
dpr
oduc
ing
an e
quiv
alen
t for
m o
f the
expr
essi
on re
veal
s so
met
hing
abo
utth
e si
tuat
ion.
ii)Ta
sks
are
limite
d to
exp
onen
tial
expr
essi
ons
with
inte
ger e
xpon
ents
.
A1.A
.SSE
.B.3
Cho
ose
and
prod
uce
an e
quiv
alen
t for
m o
f an
expr
essi
on
to re
veal
and
exp
lain
pro
perti
es o
f th
e qu
antit
y re
pres
ente
d by
the
expr
essi
on.★
a.Fa
ctor
a q
uadr
atic
exp
ress
ion
tore
veal
the
zero
s of
the
func
tion
itde
fines
.
b.C
ompl
ete
the
squa
re in
aqu
adra
tic e
xpre
ssio
n in
the
form
Ax2
+Bx
+ C
whe
reA
= 1
to re
veal
the
max
imum
or
min
imum
val
ue o
f the
func
tion
itde
fines
.
For A
1.A.
SSE.
B.3c
:
For
exa
mpl
e, th
e gr
owth
of b
acte
ria
can
be m
odel
ed b
y ei
ther
f(t)
= 3(t+
2) o
r g
(t) =
9(3
t ) b
ecau
se th
e ex
pres
sion
3
(t+2)
can
be
rew
ritte
n as
(3
t )(3
2 ) =
9(3
t ).
i)Ta
sks
have
a re
al-w
orld
con
text
. As
desc
ribed
in th
e st
anda
rd, t
here
is a
nin
terp
lay
betw
een
the
mat
hem
atic
alst
ruct
ure
of th
e ex
pres
sion
and
the
stru
ctur
e of
the
situ
atio
n su
ch th
atch
oosi
ng a
nd p
rodu
cing
an
equi
vale
ntfo
rm o
f the
exp
ress
ion
reve
als
som
ethi
ng a
bout
the
situ
atio
n.
49
c.Use the properties of
exponents to transformexpressions for exponentialfunctions. For exam
ple theexpression 1.15
t can berew
ritten as (1.151/12) 12t ≈
1.01212t to reveal the
approximate equivalent
monthly interest rate if the
annual rate is 15%.
c.Use the properties of exponents
to rewrite exponential expressions.
ii)Tasks are limited to exponential
expressions with integer exponents.
A1.A.APR.A.1 1.U
nderstand thatpolynom
ials form a system
analogous to the integers,nam
ely, they are closed underthe operations of addition,subtraction, and m
ultiplication;add, subtract, and m
ultiplypolynom
ials.
There is no additional scope or clarification inform
ation for this standard.
A1.A.APR.A.1 Understand that
polynomials form
a system
analogous to the integers, namely,
they are closed under the operations of addition, subtraction, and m
ultiplication; add, subtract, and m
ultiply polynomials.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.A.APR.B.3 3.Identify zeros ofpolynom
ials when suitable
factorizations are available,and use the zeros to constructa rough graph of the functio nd efined by the polynom
ial.
i)Tasks are limited to quadratic and
cubic polynomials in w
hich linearand quadratic factors are available.For exam
ple, find the zeros of (x -2)(x 2 - 9).
A1.A.APR.B.2 Identify zeros of polynom
ials when suitable
factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynom
ial.
Graphing is lim
ited to linear and quadratic polynom
ials
A1.A.CED.A.1 1.C
reate equations andinequalities in one variableand use them
to solveproblem
s. Include equationsarising from
linear andquadratic functions, andsim
ple rational and
i)Tasks are limited to linear,
quadratic, or exponential equationsw
ith integer exponents.
A1.A.CED.A.1 Create equations and
inequalities in one variable and use them
to solve problems.
Tasks are limited to linear, quadratic,
or exponential equations with integer
exponents.
50
expo
nent
ial f
unct
ions
.
A1.A
.CED
.A.2
2.
Cre
ate
equa
tions
in tw
o or
m
ore
varia
bles
to re
pres
ent
rela
tions
hips
bet
wee
n qu
antit
ies;
gra
ph e
quat
ions
on
coor
dina
te a
xes
with
labe
ls
and
scal
es.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd
A1.A
.CED
.A.2
Cre
ate
equa
tions
in
two
or m
ore
varia
bles
to re
pres
ent
rela
tions
hips
bet
wee
n qu
antit
ies;
gr
aph
equa
tions
with
two
varia
bles
on
coo
rdin
ate
axes
with
labe
ls a
nd
scal
es.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e.
A1.A
.CED
.A.3
3.
Rep
rese
nt c
onst
rain
ts b
y eq
uatio
ns o
r ine
qual
ities
, and
by
sys
tem
s of
equ
atio
ns
and/
or in
equa
litie
s, a
nd
inte
rpre
t sol
utio
ns a
s vi
able
or
nonv
iabl
e op
tions
in a
m
odel
ing
cont
ext.
For
exam
ple,
repr
esen
t in
equa
litie
s de
scrib
ing
nutri
tiona
l and
cos
t con
stra
ints
on
com
bina
tions
of d
iffer
ent
food
s.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd
A1.A
.CED
.A.3
Rep
rese
nt
cons
train
ts b
y eq
uatio
ns o
r in
equa
litie
s an
d by
sys
tem
s of
eq
uatio
ns a
nd/o
r ine
qual
ities
, and
in
terp
ret s
olut
ions
as
viab
le o
r no
nvia
ble
optio
ns in
a m
odel
ing
cont
ext.
For e
xam
ple,
repr
esen
t ine
qual
ities
de
scrib
ing
nutri
tiona
l and
cos
t co
nstra
ints
on
com
bina
tions
of
diffe
rent
food
s.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e.
A1.A
.CED
.A.4
4.
Rea
rrang
e fo
rmul
as to
hi
ghlig
ht a
qua
ntity
of i
nter
est,
usin
g th
e sa
me
reas
onin
g a
sin
solv
ing
equa
tions
. For
ex
ampl
e, re
arra
nge
Ohm
’s la
w
V =
IR to
hig
hlig
ht re
sist
ance
R
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd
A1.A
.CED
.A.4
Rea
rran
ge fo
rmul
as
to h
ighl
ight
a q
uant
ity o
f int
eres
t, us
ing
the
sam
e re
ason
ing
as in
so
lvin
g eq
uatio
ns.
i) Ta
sks
are
limite
d to
linea
r, qu
adra
tic,
piec
ewis
e, a
bsol
ute
valu
e, a
nd
expo
nent
ial e
quat
ions
with
inte
ger
expo
nent
s . i
i) Ta
sks
have
a re
al-w
orld
con
text
.
A1.A
.REI
.A.1
1.
Exp
lain
eac
h st
ep in
sol
ving
a
sim
ple
equa
tion
as fo
llow
ing
from
the
equa
lity
of n
umbe
rs
asse
rted
at th
e pr
evio
us s
tep,
st
artin
g fro
m th
e as
sum
ptio
n th
at th
e or
igin
al e
quat
ion
has
a so
lutio
n. C
onst
ruct
a v
iabl
e ar
gum
ent t
o ju
stify
a s
olut
ion
met
hod.
i) Ta
sks
are
limite
d to
qua
drat
ic
equa
tions
A1
.A.R
EI.A
.1 E
xpla
in e
ach
step
in
solv
ing
an e
quat
ion
as fo
llow
ing
from
the
equa
lity
of n
umbe
rs
asse
rted
at th
e pr
evio
us s
tep,
st
artin
g fro
m th
e as
sum
ptio
n th
at th
e or
igin
al e
quat
ion
has
a so
lutio
n.
Con
stru
ct a
via
ble
argu
men
t to
just
ify a
sol
utio
n m
etho
d.
Task
s ar
e lim
ited
to li
near
, qua
drat
ic,
piec
ewis
e, a
bsol
ute
valu
e, a
nd
expo
nent
ial e
quat
ions
with
inte
ger
expo
nent
s.
51
A1.A.REI.B.3 3. Solve linear equations and inequalities in one variable, including equations w
ith coefficients represented by letters.
There is no additional scope or clarification inform
ation for this standard.
A1.A.REI.B.2 Solve linear equations and inequalities in one variable, including equations w
ith coefficients represented by letters.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.A.REI.B.4 4. Solve quadratic equations in one variable. a. U
se 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 sam
e solutions. Derive
the quadratic formula from
this form
. b. Solve quadratic equations by inspection (e.g., for x
2 = 49), taking square roots, com
pleting the square, the quadratic form
ula and factoring, as appropriate to the initial form
of the equation. R
ecognize when the quadratic
formula gives com
plex solutions and w
rite them as a
± bi for real numbers a and b.
For A-REI.4b:
i) Tasks do not require students to w
rite solutions for quadratic equations that have roots w
ith nonzero im
aginary parts. How
ever, tasks can require the student to recognize cases in w
hich a quadratic equation has no real solutions. N
ote, solving a quadratic equation by factoring relies on the connection betw
een zeros and factors of polynom
ials (cluster A-APR.B).
Cluster AAPR
.B is formally
assessed in A2.
A1.A.REI.B.3 Solve quadratic equations and inequalities in one variable. a. U
se the method of com
pleting the square to rew
rite 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.
b. Solve quadratic equations by inspection (e.g., for x
2 = 49), taking square roots, com
pleting the square, know
ing and applying the quadratic form
ula, and factoring, as appropriate to the initial form
of the equation. R
ecognize when the
quadratic formula gives com
plex solutions.
For A1.A.REI.B.3b:
Tasks do not require students to write
solutions for quadratic equations that have roots w
ith nonzero imaginary
parts. How
ever, tasks can require the student to recognize cases in w
hich a quadratic equation has no real solutions. N
ote: solving a quadratic equation by factoring relies on the connection betw
een zeros and factors of polynom
ials. This is formally assessed
in Algebra II.
A1.A.REI.C.4 W
rite and solve a system
of linear equations in context.
Solve systems both algebraically and
graphically. System
s are limited to at m
ost two
equations in two variables.
A1.A.REI.C.5 5. Prove that, given a system
of tw
o equations in two
variables, replacing one equation by the sum
of that equation and a m
ultiple of the other produces a system
with
the same solutions.
There is no additional scope or clarification inform
ation for this standard.
52
A1.A
.REI
.C.6
6.
Sol
ve s
yste
ms
of li
near
eq
uatio
ns e
xact
ly a
nd
appr
oxim
atel
y (e
.g.,
with
gr
aphs
), fo
cusi
ng o
n pa
irs o
f lin
ear e
quat
ions
in tw
o va
riabl
es.
i) Ta
sks
have
a re
al-w
orld
con
text
. ii)
Tas
ks h
ave
hallm
arks
of
mod
elin
g as
a m
athe
mat
ical
pr
actic
e (le
ss d
efin
ed ta
sks,
mor
e of
th
e m
odel
ing
cycl
e, e
tc.).
A1.A
.REI
.D.1
0 10
. Und
erst
and
that
the
grap
h of
an
equa
tion
in tw
o va
riabl
es
is th
e se
t of a
ll its
sol
utio
ns
plot
ted
in th
e co
ordi
nate
pla
ne,
ofte
n fo
rmin
g a
curv
e (w
hich
co
uld
be a
line
).
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.A
.REI
.D.5
Und
erst
and
that
the
grap
h of
an
equa
tion
in tw
o va
riabl
es
is th
e se
t of a
ll its
sol
utio
ns p
lotte
d in
th
e co
ordi
nate
pla
ne, o
ften
form
ing
a cu
rve
(whi
ch c
ould
be
a lin
e).
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e
A1.A
.REI
.D.1
1 11
. Exp
lain
why
the
x-co
ordi
nate
s of
the
poin
ts
whe
re th
e gr
aphs
of t
he
equa
tions
y =
f(x)
and
y =
g(x
) in
ters
ect a
re th
e so
lutio
ns o
f th
e eq
uatio
n f(x
) = g
(x);
find
the
solu
tions
app
roxi
mat
ely,
e.
g., u
sing
tech
nolo
gy to
gra
ph
the
func
tions
, mak
e ta
bles
of
valu
es, o
r fin
d su
cces
sive
ap
prox
imat
ions
. Inc
lude
cas
es
whe
re f(
x) a
nd/o
r g(x
) are
lin
ear,
poly
nom
ial,
ratio
nal,
abso
lute
val
ue, e
xpon
entia
l, an
d lo
garit
hmic
func
tions
. ★
i) Ta
sks
that
ass
ess
conc
eptu
al
unde
rsta
ndin
g of
the
indi
cate
d co
ncep
t may
invo
lve
any
of th
e fu
nctio
n ty
pes
men
tione
d in
the
stan
dard
exc
ept e
xpon
entia
l and
lo
garit
hmic
func
tions
. ii)
Fin
ding
the
solu
tions
ap
prox
imat
ely
is li
mite
d to
cas
es
whe
re f(
x) a
nd g
(x) a
re p
olyn
omia
l fu
nctio
ns.
A1.A
.REI
.D.6
Exp
lain
why
the
x-co
ordi
nate
s of
the
poin
ts w
here
the
grap
hs o
f the
equ
atio
ns y
= f(
x) a
nd
y =
g(x)
inte
rsec
t are
the
solu
tions
of
the
equa
tion
f(x) =
g(x
); fin
d th
e ap
prox
imat
e so
lutio
ns u
sing
te
chno
logy
. ★
Incl
ude
case
s w
here
f(x)
and
/or g
(x)
are
linea
r, qu
adra
tic, a
bsol
ute
valu
e,
and
expo
nent
ial f
unct
ions
. For
ex
ampl
e, f(
x) =
3x
+ 5
and
g(x)
= x
2 +
1.
Expo
nent
ial f
unct
ions
are
lim
ited
to
dom
ains
in th
e in
tege
rs.
A1.A
.REI
.D.1
2 12
. Gra
ph th
e so
lutio
ns to
a
linea
r ine
qual
ity in
two
varia
bles
as
a ha
lf-pl
ane
(exc
ludi
ng th
e bo
unda
ry in
the
case
of a
stri
ct in
equa
lity)
, and
gr
aph
the
solu
tion
set t
o a
syst
em o
f lin
ear i
nequ
aliti
es in
tw
o va
riabl
es a
s th
e in
ters
ectio
n of
the
corr
espo
ndin
g ha
lf-pl
anes
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.A
.REI
.D.7
Gra
ph th
e so
lutio
ns to
a
linea
r ine
qual
ity in
two
varia
bles
as
a ha
lf-pl
ane
(exc
ludi
ng th
e bo
unda
ry
in th
e ca
se o
f a s
trict
ineq
ualit
y), a
nd
grap
h th
e so
lutio
n se
t to
a sy
stem
of
linea
r ine
qual
ities
in tw
o va
riabl
es a
s th
e in
ters
ectio
n of
the
corre
spon
ding
ha
lf-pl
anes
.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e.
53
A1.F.IF.A.1 1.U
nderstand that a functionfrom
one set (called thedom
ain) to another set (calledthe range) assigns to eachelem
ent of the domain exactly
one element of the range. If f
is a function and x is a nel em
ent of its domain, then
f(x) denotes the output of fcorresponding to the input x.The graph of f is the graph ofthe equation y = f(x).
There is no additional scope or clarification inform
ation for this standard.
A1.F.IF.A.1 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 elem
ent of the range. If f is a function and x is an elem
ent 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).
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.F.IF.A.2 2.U
se function notation,evaluate functions for inputs intheir dom
ains, and interpretstatem
ents that use functio nnotation in term
s of a context.
There is no additional scope or clarification inform
ation for this standard.
A1.F.IF.A.2 Use function notation,
evaluate functions for inputs in their dom
ains, and interpret statements
that use function notation in terms of
a context.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.F.IF.A.3 3.R
ecognize that sequencesare functions, som
etimes
defined recursively, whose
domain is a subset of the
integers. For example, th e
F ibonacci sequence is definedrecursively by f(0) = f(1) = 1,f(n+1) = f(n) + f(n-1) for n ≥ 1.
i)This standard is part of the Major
work in Algebra I and w
ill beassessed accordingly.
A1.F.IF.B.4 4.For a function that m
odels arelationship betw
een two
quantities, interpret keyfeatures of graphs and tablesin term
s of the quantities, andsketch graphs show
ing keyfeatures given a verbaldescription of the relationship.Key features include:intercepts; intervals w
here th ef unction is increasing,decreasing, positive, ornegative; relative m
aximum
s
i)Tasks have a real-world context.
ii)Tasks are limited to linear
functions, quadratic functions,square root functions, cube rootfunctions, piecew
ise-definedf unctions (including step functionsand absolute value functions), andexponential functions w
ith domains
in the integers.
Com
pare note (ii) with standard F-
IF.7. The function types listed here are the sam
e as those listed in the
A1.F.IF.B.3 For a function that m
odels a relationship between tw
o quantities, interpret key features of graphs and tables in term
s of the quantities, and sketch graphs show
ing key features given a verbal description of the relationship. ★
Key features include: intercepts; intervals w
here the function is increasing, decreasing, positive, or negative; relative m
aximum
s and m
inimum
s; symm
etries; and end behavior.
i)Tasks have a real-world context.
ii)Tasks are limited to linear functions,
quadratic functions, absolute val uef unctions, and exponential functionsw
ith domains in the integers.
54
and
min
imum
s; s
ymm
etrie
s;
end
beha
vior
; and
per
iodi
city
. ★
Alge
bra
I col
umn
for s
tand
ards
F-
IF.6
and
F-IF
.9.
A1.F
.IF.B
.5
5. R
elat
e th
e do
mai
n of
a
func
tion
to it
s gr
aph
and,
w
here
app
licab
le, t
o th
e qu
antit
ativ
e re
latio
nshi
p it
desc
ribes
. For
exa
mpl
e, if
the
func
tion
h(n)
giv
es th
e nu
mbe
r of
per
son-
hour
s it
take
s to
as
sem
ble
n en
gine
s in
a
fact
ory,
then
the
posi
tive
inte
gers
wou
ld b
e an
ap
prop
riate
dom
ain
for t
he
func
tion.★
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.F
.IF.B
.4 R
elat
e th
e do
mai
n of
a
func
tion
to it
s gr
aph
and,
whe
re
appl
icab
le, t
o th
e qu
antit
ativ
e re
latio
nshi
p it
desc
ribes
. ★
For e
xam
ple,
if th
e fu
nctio
n h(
n) g
ives
th
e nu
mbe
r of p
erso
n-ho
urs
it ta
kes
to
asse
mbl
e n
engi
nes
in a
fact
ory,
then
th
e po
sitiv
e in
tege
rs w
ould
be
an
appr
opria
te d
omai
n fo
r the
func
tion.
T
here
are
no
asse
ssm
ent l
imits
for
this
sta
ndar
d. T
he e
ntire
sta
ndar
d is
as
sess
ed in
this
cou
rse.
A1.F
.IF.B
.6
6. C
alcu
late
and
inte
rpre
t the
av
erag
e ra
te o
f cha
nge
of a
fu
nctio
n (p
rese
nted
sy
mbo
lical
ly o
r as
a ta
ble)
ov
er a
spe
cifie
d in
terv
al.
Estim
ate
the
rate
of c
hang
e fro
m a
gra
ph.★
i) Ta
sks
have
a re
al-w
orld
con
text
. ii)
Tas
ks a
re lim
ited
to li
near
fu
nctio
ns, q
uadr
atic
func
tions
, sq
uare
root
func
tions
, cub
e ro
ot
func
tions
, pie
cew
ise-
defin
ed
func
tions
(inc
ludi
ng s
tep
func
tions
an
d ab
solu
te v
alue
func
tions
), an
d ex
pone
ntia
l fun
ctio
ns w
ith d
omai
ns
in th
e in
tege
rs.
The
func
tion
type
s lis
ted
here
are
th
e sa
me
as th
ose
liste
d in
the
Alge
bra
I col
umn
for s
tand
ards
F-
IF.4
and
F-IF
.9.
A1.F
.IF.B
.5 C
alcu
late
and
inte
rpre
t th
e av
erag
e ra
te o
f cha
nge
of a
fu
nctio
n (p
rese
nted
sym
bolic
ally
or
as a
tabl
e) o
ver a
spe
cifie
d in
terv
al.
Estim
ate
the
rate
of c
hang
e fro
m a
gr
aph.★
i) Ta
sks
have
a re
al-w
orld
con
text
. ii
) Tas
ks a
re li
mite
d to
line
ar fu
nctio
ns,
quad
ratic
func
tions
, pie
cew
ise-
defin
ed
func
tions
(inc
ludi
ng s
tep
func
tions
and
ab
solu
te v
alue
func
tions
), an
d ex
pone
ntia
l fun
ctio
ns w
ith d
omai
ns in
th
e in
tege
rs.
A1.F
.IF.C
.7
7. G
raph
func
tions
exp
ress
ed
sym
bolic
ally
and
sho
w k
ey
feat
ures
of t
he g
raph
, by
hand
in
sim
ple
case
s an
d us
ing
tech
nolo
gy fo
r mor
e co
mpl
icat
ed c
ases
. ★
a. G
raph lin
ear
and
quad
ratic
func
tions
and
sho
w
inte
rcep
ts, m
axim
a, a
nd
min
ima.
b
. Gra
ph s
quar
e ro
ot, c
ube
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.F
.IF.C
.6 G
raph
func
tions
ex
pres
sed
sym
bolic
ally
and
sho
w
key
feat
ures
of t
he g
raph
, by
hand
an
d us
ing
tech
nolo
gy.
a. G
raph
line
ar a
nd q
uadr
atic
fu
nctio
ns a
nd s
how
inte
rcep
ts,
max
ima,
and
min
ima.
b
. Gra
ph s
quar
e ro
ot, c
ube
root
, an
d pi
ecew
ise-
defin
ed fu
nctio
ns,
incl
udin
g st
ep fu
nctio
ns a
nd
abso
lute
val
ue fu
nctio
ns.
Task
s in
A1.
F.IF
.C.6
b ar
e lim
ited
to
piec
ewis
e, s
tep
and
abso
lute
val
ue
func
tions
.
55
root, and piecewise-defined
functions, including step functions and absolute value functions
A1.F.IF.C.8 8. W
rite a function defined by an expression in different but equivalent form
s to reveal and explain different properties of the function. a. U
se the process of factoring and com
pleting the square in a quadratic function to show
zeros, extrem
e values, and sym
metry of the graph, and
interpret these in terms of a
context.
There is no additional scope or clarification inform
ation for this standard.
A1.F.IF.C.7 Write a function defined
by an expression in different but equivalent form
s to reveal and explain different properties of the function. a. U
se the process of factoring and com
pleting the square in a quadratic function to show
zeros, extrem
e values, and sym
metry of the graph,
and interpret these in term
s of a context.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.F.IF.C.9 9. C
ompare properties of tw
o functions each represented in a different w
ay (algebraically, graphically, num
erically in tables, or by verbal descriptions). For exam
ple, given a graph of one quadratic function and an algebraic expression for another, say w
hich has the larger m
aximum
.
i) Tasks are limited to linear
functions, quadratic functions, square root functions, cube root functions, piecew
ise-defined functions (including step functions and absolute value functions), and exponential functions w
ith domains
in the integers. The function types listed here are the sam
e as those listed in the Algebra I colum
n for standards F-IF.4 and F-IF.6.
A1.F.IF.C.8 Com
pare properties of tw
o functions each represented in a different w
ay (algebraically, graphically, num
erically in tables, or by verbal descriptions).
i) Tasks have a real-world context.
ii) Tasks are limited to linear functions,
quadratic functions, piecewise-defined
functions (including step functions and absolute value functions), and exponential functions w
ith domains in
the integers.
A1.F.BF.A.1 1. W
rite a function that describes a relationship betw
een two quantities.
★ a
. Dete
rmin
e a
n e
xplicit
expression, a recursive process, or steps for calculation from
a context.
i) Tasks have a real-world context.
ii) Tasks are limited to linear
functions, quadratic functions, and exponential functions w
ith domains
in the integers.
A1.F.BF.A.1 Write a function that
describes a relationship between
two quantities.★
a. D
etermine an explicit
expression, a recursive process, or steps for calculation from
a context.
i) Tasks have a real-world context.
ii) Tasks are limited to linear functions,
quadratic functions, and exponential functions w
ith domains in the integers.
56
A1.F
.BF.
B.3
3. Id
entif
y th
e ef
fect
on
the
grap
h of
repl
acin
g f(x
) by
f(x) +
k,
k f(
x), f
(kx)
, and
f(x
+ k)
for
spec
ific
valu
es o
f k (b
oth
posi
tive
and
nega
tive)
; fin
d th
e va
lue
of k
giv
en th
e gr
aphs
. Ex
perim
ent w
ith c
ases
and
ill
ustra
te a
n ex
plan
atio
n of
the
effe
cts
on th
e gr
aph
usin
g te
chno
logy
. Inc
lude
re
cogn
izin
g ev
en a
nd o
dd
func
tion
from
thei
r gra
phs
and
alge
brai
c ex
pres
sion
s fo
r th
em.
i) Id
entif
ying
the
effe
ct o
n th
e gr
aph
of re
plac
ing
f(x) b
y f(x
) + k
, k f(
x),
f(kx)
, and
f(x+
k) fo
r spe
cific
val
ues
of k
(bot
h po
sitiv
e an
d ne
gativ
e) is
lim
ited
to li
near
and
qua
drat
ic
func
tions
. ii
) Exp
erim
entin
g w
ith c
ases
and
illu
stra
ting
an e
xpla
natio
n of
the
effe
cts
on th
e gr
aph
usin
g te
chno
logy
is li
mite
d to
linea
r fu
nctio
ns, q
uadr
atic
func
tions
, sq
uare
root
func
tions
, cub
e ro
ot
func
tions
, pie
cew
ise-
defin
ed
func
tions
(inc
ludi
ng s
tep
func
tions
an
d ab
solu
te v
alue
func
tions
), an
d ex
pone
ntia
l fun
ctio
ns w
ith d
omai
ns
in th
e in
tege
rs.
iii) T
asks
do
not i
nvol
ve re
cogn
izin
g ev
en a
nd o
dd fu
nctio
ns.
The
func
tion
type
s lis
ted
in n
ote
(ii)
are
the
sam
e as
thos
e lis
ted
in th
e Al
gebr
a I c
olum
n fo
r sta
ndar
ds F
-IF
.4, F
-IF.6
, and
F-IF
.9.
A1.F
.BF.
B.2
Iden
tify
the
effe
ct o
n th
e gr
aph
of re
plac
ing
f(x) b
y f(x
) + k
, k
f(x),
f(kx)
, and
f(x
+ k)
for s
peci
fic
valu
es o
f k (b
oth
posi
tive
and
nega
tive)
; fin
d th
e va
lue
of k
giv
en
the
grap
hs. E
xper
imen
t with
cas
es
and
illus
trate
an
expl
anat
ion
of th
e ef
fect
s on
the
grap
h us
ing
tech
nolo
gy.
i) Id
entif
ying
the
effe
ct o
n th
e gr
aph
of
repl
acin
g f(x
) by
f(x) +
k, k
f(x)
, and
f(x
+k) f
or s
peci
fic v
alue
s of
k (b
oth
posi
tive
and
nega
tive)
is li
mite
d to
lin
ear,
quad
ratic
, and
abs
olut
e va
lue
func
tions
. ii
) f(k
x) w
ill no
t be
incl
uded
in A
lgeb
ra
1. It
is a
ddre
ssed
in A
lgeb
ra 2
. ii
i) Ex
perim
entin
g w
ith c
ases
and
illu
stra
ting
an e
xpla
natio
n of
the
effe
cts
on th
e gr
aph
usin
g te
chno
logy
is
limite
d to
line
ar fu
nctio
ns, q
uadr
atic
fu
nctio
ns, a
bsol
ute
valu
e, a
nd
expo
nent
ial f
unct
ions
with
dom
ains
in
the
inte
gers
. iv
) Tas
ks d
o no
t inv
olve
reco
gniz
ing
even
and
odd
func
tions
.
A1.F
.LE.
A.1
1. D
istin
guis
h be
twee
n si
tuat
ions
that
can
be
mod
eled
w
ith li
near
func
tions
and
with
ex
pone
ntia
l fun
ctio
ns.
a. P
rove
that
line
ar fu
nctio
ns
grow
by
equa
l diff
eren
ces
over
eq
ual i
nter
vals
, and
that
ex
pone
ntia
l fun
ctio
ns g
row
by
equa
l fac
tors
ove
r equ
al
inte
rval
s.
b. R
ecog
nize
situ
atio
ns in
w
hich
one
qua
ntity
cha
nges
at
a co
nsta
nt ra
te p
er u
nit
inte
rval
rela
tive
to a
noth
er.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.F
.LE.
A.1
Dis
tingu
ish
betw
een
situ
atio
ns th
at c
an b
e m
odel
ed w
ith
linea
r fun
ctio
ns a
nd w
ith e
xpon
entia
l fu
nctio
ns.
a. R
ecog
nize
that
line
ar fu
nctio
ns
grow
by
equa
l diff
eren
ces
over
equ
al
inte
rval
s an
d th
at e
xpon
entia
l fu
nctio
ns g
row
by
equa
l fac
tors
ove
r eq
ual i
nter
vals
. b.
Rec
ogni
ze s
ituat
ions
in w
hich
one
qu
antit
y ch
ange
s at
a c
onst
ant r
ate
per u
nit i
nter
val r
elat
ive
to a
noth
er.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e.
57
c. Recognize situations in
which a quantity grow
s or decays by a constant percent rate per unit interval relative to another.
c. Recognize situations in w
hich a quantity grow
s or decays by a constant factor per unit interval relative to another.
A1.F.LE.A.2 2. C
onstruct linear and exponential functions, including arithm
etic and geom
etric sequences, given a graph, a description of a relationship, or tw
o input-output pairs (include reading these from
a table).
i) Tasks are limited to constructing
linear and exponential functions in sim
ple context (not multi-step).
A1.F.LE.A.2 Construct linear and
exponential functions, including arithm
etic and geometric sequences,
given a graph, a table, a description of a relationship, or input-output pairs.
Tasks are limited to constructing linear
and exponential functions in simple
context (not multi-step).
A1.F.LE.A.3 3. O
bserve using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (m
ore generally) as a polynom
ial function.
There is no additional scope or clarification inform
ation for this standard.
A1.F.LE.A.3 Observe using graphs
and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (m
ore generally) as a polynom
ial function.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.F.LE.B.5 5. Interpret the param
eters in a linear or exponential function in term
s of a context.
i) Tasks have a real-world context.
ii) Exponential functions are limited
to those with dom
ains in the integers.
F.LE.B.4 Interpret the parameters in
a linear or exponential function in term
s of a context.
For example, the total cost of an
electrician who charges 35 dollars for a
house call and 50 dollars per hour w
ould be expressed as the function y = 50x + 35. If the rate w
ere raised to 65 dollars per hour, describe how
the function w
ould change. i) Tasks have a real-w
orld context. ii) Exponential functions are lim
ited to those w
ith domains in the integers.
58
A1.S
.ID.A
.1
1. R
epre
sent
dat
a w
ith p
lots
on
the
real
num
ber l
ine
(dot
pl
ots,
his
togr
ams,
and
box
pl
ots)
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.S
.ID.A
.1 R
epre
sent
sin
gle
or
mul
tiple
dat
a se
ts w
ith d
ot p
lots
, hi
stog
ram
s, s
tem
plo
ts (s
tem
and
le
af),
and
box
plot
s.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e.
A1.S
.ID.A
.2
2. U
se s
tatis
tics
appr
opria
te to
th
e sh
ape
of th
e da
ta
dist
ribut
ion
to c
ompa
re c
ente
r (m
edia
n, m
ean)
and
spr
ead
(inte
rqua
rtile
rang
e, s
tand
ard
devi
atio
n) o
f tw
o or
mor
e di
ffere
nt d
ata
sets
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.S
.ID.A
.2 U
se s
tatis
tics
appr
opria
te to
the
shap
e of
the
data
di
strib
utio
n to
com
pare
cen
ter
(med
ian,
mea
n) a
nd s
prea
d (in
terq
uarti
le ra
nge,
sta
ndar
d de
viat
ion)
of t
wo
or m
ore
diffe
rent
da
ta s
ets.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e.
A1.S
.ID.A
.3
3. In
terp
ret d
iffer
ence
s in
sh
ape,
cen
ter,
and
spre
ad in
th
e co
ntex
t of t
he d
ata
sets
, ac
coun
ting
for p
ossi
ble
effe
cts
of e
xtre
me
data
poi
nts
(out
liers
).
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.S
.ID.A
.3 In
terp
ret d
iffer
ence
s in
sh
ape,
cen
ter,
and
spre
ad in
the
cont
ext o
f the
dat
a se
ts, a
ccou
ntin
g fo
r pos
sibl
e ef
fect
s of
ext
rem
e da
ta
poin
ts (o
utlie
rs).
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is
asse
ssed
in th
is c
ours
e.
A1.S
.ID.B
.5
5. S
umm
ariz
e ca
tego
rical
dat
a fo
r tw
o ca
tego
ries
in tw
o-w
ay
frequ
ency
tabl
es. I
nter
pret
re
lativ
e fre
quen
cies
in th
e co
ntex
t of t
he d
ata
(incl
udin
g jo
int,
mar
gina
l, an
d co
nditi
onal
re
lativ
e fre
quen
cies
). R
ecog
nize
pos
sibl
e as
soci
atio
ns a
nd tr
ends
in th
e da
ta
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arific
atio
n in
form
atio
n fo
r thi
s st
anda
rd.
A1.S
.ID.B
.6
6. R
epre
sent
dat
a on
two
quan
titat
ive
varia
bles
on
a sc
atte
r plo
t, an
d de
scrib
e ho
w
the
varia
bles
are
rela
ted.
a.
Fit
a fu
nctio
n to
the
data
; us
e fu
nctio
ns fi
tted
to d
ata
to
solv
e pr
oble
ms
in th
e co
ntex
t of
the
data
. Use
giv
en
func
tions
or c
hoos
e a
func
tion
sugg
este
d by
the
cont
ext.
For S
-ID.6
a:
i) T
asks
hav
e a
real
-wor
ld c
onte
xt.
ii) E
xpon
entia
l fun
ctio
ns a
re li
mite
d to
thos
e w
ith d
omai
ns in
the
inte
gers
.
A1.S
.ID.B
.4 R
epre
sent
dat
a on
two
quan
titat
ive
varia
bles
on
a sc
atte
r pl
ot, a
nd d
escr
ibe
how
the
varia
bles
ar
e re
late
d.
a. F
it a
func
tion
to th
e da
ta; u
se
func
tions
fitte
d to
dat
a to
sol
ve
prob
lem
s in
the
cont
ext o
f the
dat
a.
Use
giv
en fu
nctio
ns o
r cho
ose
a fu
nctio
n su
gges
ted
by th
e co
ntex
t.
Emph
asiz
e lin
ear m
odel
s, q
uadr
atic
m
odel
s, a
nd e
xpon
entia
l mod
els
with
do
mai
ns in
the
inte
gers
. F
or A
1.S.
ID.B
.4a:
i)
Tas
ks h
ave
a re
al-w
orld
con
text
. ii
) Exp
onen
tial f
unct
ions
are
lim
ited
to
thos
e w
ith d
omai
ns in
the
inte
gers
.
59
Emphasize linear, quadratic,
and exponential models.
b. Informally assess the fit of
a function by plotting andanalyzing residuals.
c.Fit a linear function for ascatter plot that suggests alinear association.
b.Fit a linear function for a scatterplot that suggests a linearassociation.
A1.S.ID.C.7 7.Interpret the slope (rate ofchange) and the intercept(constant term
) of a linearm
odel in the context of th edata.
There is no additional scope or clarification inform
ation for this standard.
A1.S.ID.C.5 Interpret the slope (rate of change) and the intercept (constant term
) of a linear model in
the context of the data.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.S.ID.C.8 8.C
ompute (using technology)
and interpret the correlationcoefficient of a linear fit.
There is no additional scope or clarification inform
ation for this standard.
A1.S.ID.C.6 Use technology to
compute and interpret the correlation
coefficient of a linear fit.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A1.S.ID.C.9 9.D
istinguish between
correlation and causation.There is no additional scope or clarification inform
ation for this standard.
A1.S.ID.C.7 Distinguish betw
een correlation and causation.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
Major W
ork of the Grade
Supporting Work
60
Not
atio
n TN
Sta
ndar
ds th
roug
h M
ay 2
017
Stan
dard
Cla
rific
atio
n Re
vise
d TN
Sta
ndar
ds
Stan
dard
Cla
rifi
cati
on
A2.N
.RN.
A.1
1.
Exp
lain
how
the
defin
ition
of
the
mea
ning
of r
atio
nal
expo
nent
s fo
llow
s fro
m
exte
ndin
g th
e pr
oper
ties
of
inte
ger e
xpon
ents
to th
ose
valu
es, a
llow
ing
for a
not
atio
n fo
r rad
ical
s in
term
s of
ratio
nal
expo
nent
s. F
or e
xam
ple,
we
defin
e 51/
3 to
be
the
cube
root
of
5 b
ecau
se w
e w
ant (
51/3 )3 =
5 (
1/3 )3 t
o ho
ld, s
o (5
1/3 )3 m
ust
equa
l 5.
Th
ere
is n
o ad
ditio
nal s
cope
or
clar
ifica
tion
info
rmat
ion
for t
his
stan
dard
.
A2.N
.RN.
A.1
Expl
ain
how
the
defin
ition
of t
he m
eani
ng o
f rat
iona
l ex
pone
nts
follo
ws
from
ext
endi
ng
the
prop
ertie
s of
inte
ger e
xpon
ents
to
thos
e va
lues
, allo
win
g fo
r a
nota
tion
for r
adic
als
in te
rms
of
ratio
nal e
xpon
ents
.
For e
xam
ple,
we
defin
e 51/
3 to
be th
e cu
be ro
ot
of 5
bec
ause
we
wan
t (51/
3 )3 =
5(⅓
)3 to
hol
d, s
o (5
1/3 )
3 mus
t equ
al 5
. Th
ere
are
no a
sses
smen
t lim
its fo
r thi
s st
anda
rd. T
he e
ntire
sta
ndar
d is
ass
esse
d in
th
is c
ours
e.
A2.N
.RN.
A.2
2.
Rew
rite
expr
essi
ons
invo
lvin
g ra
dica
ls a
nd ra
tiona
l ex
pone
nts
usin
g th
e pr
oper
ties
of e
xpon
ents
.
Th
ere
is n
o ad
ditio
nal s
cope
or
clar
ifica
tion
info
rmat
ion
for t
his
stan
dard
.
A2.N
.RN.
A.2
Rew
rite
expr
essi
ons
invo
lvin
g ra
dica
ls a
nd ra
tiona
l ex
pone
nts
usin
g th
e pr
oper
ties
of
expo
nent
s.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is a
sses
sed
in
this
cou
rse.
A2.N
.NQ
.A.2
2.
Def
ine
appr
opria
te
quan
titie
s fo
r the
pur
pose
of
desc
riptiv
e m
odel
ing.
This
sta
ndar
d w
ill be
ass
esse
d in
A
lgeb
ra II
by
ensu
ring
that
som
e m
odel
ing
task
s (in
volv
ing
Alg
ebra
II
cont
ent o
r sec
urel
y he
ld c
onte
nt
from
pre
viou
s gr
ades
and
co
urse
s) re
quire
the
stud
ent t
o cr
eate
a q
uant
ity o
f int
eres
t in
the
situ
atio
n be
ing
desc
ribed
(i.e
., th
is
is n
ot p
rovi
ded
in th
e ta
sk).
For
exam
ple,
in a
situ
atio
n in
volv
ing
perio
dic
phen
omen
a, th
e st
uden
t m
ight
aut
onom
ousl
y de
cide
that
am
plitu
de is
a k
ey v
aria
ble
in a
si
tuat
ion,
and
then
cho
ose
to w
ork
with
pea
k am
plitu
de.
A2.N
Q.A
.1 Id
entif
y, in
terp
ret,
and
just
ify a
ppro
pria
te q
uant
ities
for t
he
purp
ose
of d
escr
iptiv
e m
odel
ing.
Des
crip
tive
mod
elin
g re
fers
to u
nder
stan
ding
an
d in
terp
retin
g gr
aphs
; ide
ntify
ing
extra
neou
s in
form
atio
n; c
hoos
ing
appr
opria
te u
nits
; etc
. Th
ere
are
no a
sses
smen
t lim
its fo
r thi
s st
anda
rd. T
he e
ntire
sta
ndar
d is
ass
esse
d in
th
is c
ours
e.
A2.N
.CN.
A.1
1. K
now
ther
e is
a c
ompl
ex
num
ber i
suc
h th
at i2 =
–1,
and
ev
ery
com
plex
num
ber h
as th
e fo
rm a
+ b
i with
a a
nd b
real
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n in
form
atio
n fo
r thi
s st
anda
rd.
A2.N
.CN.
A.1
Know
ther
e is
a
com
plex
num
ber i
suc
h th
at i2 =
–1,
an
d ev
ery
com
plex
num
ber h
as th
e fo
rm a
+ b
i with
a a
nd b
real
.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire s
tand
ard
is a
sses
sed
in
this
cou
rse.
61
A2.N.CN.A.2 2. U
se the relation i 2 = –1 and the com
mutative, associative,
and distributive properties to add, subtract, and m
ultiply com
plex numbers.
There is no additional scope or clarification inform
ation for this standard.
A2.N.CN.A.2 Know and use the
relation i 2= –1 and the com
mutative, associative, and
distributive properties to add, subtract, and m
ultiply complex
numbers.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A2.N.CN.B.7 7. Solve quadratic equations w
ith real coefficients that have com
plex solutions.
There is no additional scope or clarification inform
ation for this standard.
A2.N.CN.B.3 Solve quadratic equations w
ith real coefficients that have com
plex solutions.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A2.A.SSE.A.2 2. U
se the structure of an expression to identify w
ays to rew
rite 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).
i) Tasks are limited to polynom
ial, rational, or exponential expressions. ii) E
xamples: see x
4 – y4 as (x
2 ) 2 – (y 2) 2 , thus recognizing it as a difference of squares that can be factored as (x
2 – y2 )(x
2 + y2 ). In
the equation x2 + 2x + 1 + y
2 = 9, see an opportunity to rew
rite the first three term
s as (x+1) 2 , thus recognizing the equation of a circle w
ith radius 3 and center (-1, 0). S
ee (x2 + 4)/(x
2 + 3) as ( (x 2+3) + 1 )/(x
2 +3), thus recognizing an opportunity to w
rite it as 1 + 1/(x
2 +3).
A2.A.SSE.A.1 Use the structure of
an expression to identify ways to
rewrite it.
For example, see 2x
4 + 3x2 – 5 as its factors (x
2 – 1) and (2x2 + 5); see x
4 – y4 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 ); see (x 2 + 4)/(x 2 + 3) as ((x 2+ 3) + 1 )/(x
2 + 3), thus recognizing an opportunity to w
rite it as 1 + 1/(x 2 + 3). Tasks are lim
ited to polynom
ial, rational, or exponential expressions.
A2.A.SSE.B.3 3. C
hoose and produce an equivalent form
of an expression to reveal and explain properties of the quantity represented by the expression.★
c. U
se the properties of exponents to transform
expressions for exponential
i) Tasks have a real-world context.
As described in the standard,
there is an interplay between the
mathem
atical structure of the expression and the structure of the situation such that choosing and producing an equivalent form
of the expression reveals som
ething about the situation. ii) Tasks are lim
ited to exponential
A2.A.SSE.B.2 Choose and
produce an equivalent form of an
expression to reveal and explain properties of the quantity represented by the expression.★
a. U
se the properties of exponents to rew
rite expressions for exponential functions.
For example the expression 1.15
t can be rewritten as
((1.15) 1/12) 12t ≈ 1.01212t to reveal that the
approximate equivalent m
onthly interest rate is 1.2%
if the annual rate is 15%.
i) Tasks have a real-world context. A
s described in the standard, there is an interplay betw
een the m
athematical structure of the expression and the
structure of the situation such that choosing and producing an equivalent form
of the expression
62
func
tions
. For
exa
mpl
e th
e ex
pres
sion
1.15
t can
be
rew
ritte
n as
(1.1
51/12
)12t ≈
1.01
212t t
o re
veal
the
appr
oxim
ate
equi
vale
nt m
onth
ly in
tere
st ra
te
if th
e an
nual
rate
is 1
5%.
expr
essi
ons
with
ratio
nal o
r rea
l ex
pone
nts.
re
veal
s so
met
hing
abo
ut th
e si
tuat
ion.
i i)Ta
sks
are
limite
d to
exp
onen
tial e
xpre
ssio
ns w
ithra
tiona
l or r
eal e
xpon
ents
.
A2.A
.SSE
.B.4
4.
Der
ive
the
form
ula
for t
hesu
m o
f a fi
nite
geo
met
ricse
ries
(whe
n th
e co
mm
on ra
ti ois
not
1),
and
use
the
form
ula
to s
olve
pro
blem
s. F
or e
xam
ple,
calcu
late
mor
tgag
e pa
ymen
ts.★
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n in
form
atio
n fo
r thi
s st
anda
rd.
A2.A
.SSE
.B.3
Rec
ogni
ze a
fini
te
geom
etric
ser
ies
(whe
n th
e co
mm
on ra
tio is
not
1),
and
know
an
d us
e th
e su
m fo
rmul
a to
sol
ve
prob
lem
s in
con
text
.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is c
ours
e.
A2.A
.APR
.A.2
2.
Know
and
app
ly th
eR
emai
nder
The
orem
: For
apo
lyno
mia
l p(x
) and
a n
umbe
ra,
the
rem
aind
er o
n di
visi
on b
yx
– a
is p
(a),
so p
(a) =
0 if
and
only
if (x
– a
) is
a fa
ctor
of p
(x).
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n in
form
atio
n fo
r thi
s st
anda
rd.
A2.A
.APR
.A.1
Kno
w a
nd a
pply
the
Rem
aind
er T
heor
em: F
or a
po
lyno
mia
l p(x
) and
a n
umbe
r a,
the
rem
aind
er o
n di
visi
on b
y x
– a
is p
(a),
so p
(a) =
0 if
and
onl
y if
(x –
a)
is a
fact
or o
f p(x
).
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is c
ours
e.
A2.A
.APR
.A.3
3.
Iden
tify
zero
s of
poly
nom
ials
whe
n su
itabl
ef a
ctor
izat
ions
are
ava
ilabl
e,an
d us
e th
e ze
ros
to c
onst
ruct
a ro
ugh
grap
h of
the
func
tion
defin
ed b
y th
e po
lyno
mia
l.
i)Ta
sks
incl
ude
quad
ratic
, cub
ic,
and
quar
tic p
olyn
omia
ls a
ndpo
lyno
mia
ls fo
r whi
ch fa
ctor
s ar
eno
t pro
vide
d. F
or e
xam
ple,
find
the
zero
s of
(x2 -
1)(
x2 + 1
)
A2.A
.APR
.A.2
Iden
tify
zero
s of
po
lyno
mia
ls w
hen
suita
ble
fact
oriz
atio
ns a
re a
vaila
ble,
and
us
e th
e ze
ros
to c
onst
ruct
a ro
ugh
grap
h of
the
func
tion
defin
ed b
y th
e po
lyno
mia
l.
Task
s in
clud
e qu
adra
tic, c
ubic
, and
qua
rtic
poly
nom
ials
and
pol
ynom
ials
for w
hich
fact
ors
are
not p
rovi
ded.
For
exa
mpl
e, fi
nd th
e ze
ros
of
(x 2
- 1)(
x 2 +
1).
A2.A
.APR
.B.4
4.
Prov
e po
lyno
mia
l ide
ntiti
esan
d us
e th
em to
des
crib
enu
mer
ical
rela
tions
hips
. For
ex
ampl
e, th
e po
lyno
mia
l ide
ntity
(x
2 +
y2 ) 2 =
(x2 –
y2 )
2 +(2
xy) 2
can
be u
sed
to g
ener
ate
Pyth
agor
ean
trip
les.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n in
form
atio
n fo
r thi
s st
anda
rd.
A2.A
.APR
.B.3
Kno
w a
nd u
se
poly
nom
ial i
dent
ities
to d
escr
ibe
num
eric
al re
latio
nshi
ps.
For e
xam
ple,
com
pare
(31)
(29)
= (3
0 +
1) (3
0 –
1) =
30
2 –
1 2
with
(x +
y) (
x –
y) =
x2 –
y 2 .
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is c
ours
e.
A2.A
.APR
.C.6
6.
Rew
rite
sim
ple
ratio
nal
expr
essi
ons
in d
iffer
ent f
orm
s;w
rite
a(x)
/b(x
) in
the
form
q(x
) +r(
x)/b
(x),
whe
re a
(x),
b(x)
, q(x
),an
d r(x
) are
pol
ynom
ials
with
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n in
form
atio
n fo
r thi
s st
anda
rd.
A2.A
.APR
.C.4
Rew
rite
ratio
nal
expr
essi
ons
in d
iffer
ent f
orm
s.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is c
ours
e.
63
the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the m
ore complicated
examples, a com
puter algebra system
.
A2.A.CED.A.1 1. C
reate equations and inequalities in one variable and use them
to solve problem
s. Include equations arising from
linear and quadratic functions, and sim
ple rational and exponential functions.
i) Tasks are limited to exponential
equations with rational or real
exponents and rational functions. ii) Tasks have a real-w
orld context.
A2.A.CED.A.1 Create equations
and inequalities in one variable and use them
to solve problems.
Include equations arising from linear and quadratic
functions, and rational and exponential functions. Tasks have a real-w
orld context.
A2.A.CED.A.2 R
earrange formulas
to highlight a quantity of interest, using the sam
e reasoning as in solving equations.
i) Tasks are limited to square root, cube root,
polynomial, rational, and logarithm
ic functions. ii) Tasks have a real-w
orld context
A2.A.REI.A.1 1. Explain each step in solving a sim
ple equation as following
from the equality of num
bers asserted at the previous step, starting from
the assumption
that the original equation has a solution. C
onstruct a viable argum
ent to justify a solution m
ethod.
i) Tasks are limited to sim
ple rational or radical equations
A2.A.REI.A.1 Explain each step in solving an equation as follow
ing from
the equality of numbers
asserted at the previous step, starting from
the assumption that
the original equation has a solution. C
onstruct a viable argument to
justify a solution method.
Tasks are limited to square root, cube root,
polynomial, rational, and logarithm
ic functions.
A2.A.REI.A.2 2. Solve sim
ple rational and radical equations in one variable, and give exam
ples show
ing how extraneous
solutions may arise.
There is no additional scope or clarification inform
ation for this standard.
A2.A.REI.A.2 Solve rational and radical equations in one variable, and identify extraneous solutions w
hen they exist.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A2.A.REI.B.4 4. Solve quadratic equations in
i) In the case of equations that have roots w
ith nonzero imaginary
A2.A.REI.B.3 Solve quadratic In the case of equations that have roots w
ith nonzero im
aginary parts, students write the solutions as a ±
64
one
varia
ble.
b.
Sol
ve q
uadr
atic
equ
atio
ns
by in
spec
tion
(e.g
., fo
r x2 =
49
), ta
king
squ
are
root
s,
com
plet
ing
the
squa
re, t
he
quad
ratic
form
ula
and
fact
orin
g, a
s ap
prop
riate
to th
e in
itial
form
of t
he e
quat
ion.
R
ecog
nize
whe
n th
e qu
adra
tic
form
ula
give
s co
mpl
ex
solu
tions
and
writ
e th
em a
s a
± bi
for r
eal n
umbe
rs a
and
b.
parts
, stu
dent
s w
rite
the
solu
tions
as
a ±
bi f
or re
al n
umbe
rs
a an
d b.
equa
tions
and
ineq
ualit
ies
in o
ne
varia
ble.
a.
Sol
ve q
uadr
atic
equ
atio
ns b
y in
spec
tion
(e.g
., fo
r x 2 =
49)
, tak
ing
squa
re ro
ots,
com
plet
ing
the
squa
re, k
now
ing
and
appl
ying
the
quad
ratic
form
ula,
and
fact
orin
g, a
s ap
prop
riate
to th
e in
itial
form
of t
he
equa
tion.
Rec
ogni
ze w
hen
the
quad
ratic
form
ula
give
s co
mpl
ex
solu
tions
and
writ
e th
em a
s a
± bi
fo
r rea
l num
bers
a a
nd b
.
bi fo
r rea
l num
bers
a a
nd b
A2.A
.REI
.C.6
6.
Sol
ve s
yste
ms
of li
near
eq
uatio
ns e
xact
ly a
nd
appr
oxim
atel
y (e
.g.,
with
gr
aphs
), fo
cusi
ng o
n pa
irs o
f lin
ear e
quat
ions
in tw
o va
riabl
es.
i) Ta
sks
are
limite
d to
3x3
sy
stem
s.
A2.A
.REI
.C.4
Writ
e an
d so
lve
a sy
stem
of l
inea
r equ
atio
ns in
co
ntex
t.
Whe
n so
lvin
g al
gebr
aica
lly, t
asks
are
lim
ited
to
syst
ems
of a
t mos
t thr
ee e
quat
ions
and
thre
e va
riabl
es. W
ith g
raph
ic s
olut
ions
, sys
tem
s ar
e lim
ited
to o
nly
two
varia
bles
.
A2.A
.REI
.C.7
7.
Sol
ve a
sim
ple
syst
em
cons
istin
g of
a li
near
equ
atio
n an
d a
quad
ratic
equ
atio
n in
tw
o va
riabl
es a
lgeb
raic
ally
and
gr
aphi
cally
. For
exa
mpl
e, fi
nd
the
poin
ts o
f int
erse
ctio
n be
twee
n th
e lin
e y
= –3
x and
the
circ
le x
2 + y
2 = 3
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n in
form
atio
n fo
r thi
s st
anda
rd.
A2.A
.REI
.C.5
Sol
ve a
sys
tem
co
nsis
ting
of a
line
ar e
quat
ion
and
a qu
adra
tic e
quat
ion
in tw
o va
riabl
es a
lgeb
raic
ally
and
gr
aphi
cally
.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is c
ours
e.
A2.A
.REI
.D.1
1 11
. Exp
lain
why
the
x-co
ordi
nate
s of
the
poin
ts
whe
re th
e gr
aphs
of t
he
equa
tions
y
= f(x
) and
y =
g(x
) int
erse
ct
are
the
solu
tions
of t
he
equa
tion
f(x) =
g(x
); fin
d th
e so
lutio
ns a
ppro
xim
atel
y, e
.g.,
usin
g te
chno
logy
to g
raph
the
func
tions
, mak
e ta
bles
of
valu
es, o
r fin
d su
cces
sive
ap
prox
imat
ions
. Inc
lude
cas
es
i) Ta
sks
may
invo
lve
any
of th
e fu
nctio
n ty
pes
men
tione
d in
the
stan
dard
.
A2.A
.REI
.D.6
Exp
lain
why
the
x-co
ordi
nate
s of
the
poin
ts w
here
the
grap
hs o
f the
equ
atio
ns y
= f(
x) a
nd
y =
g(x)
inte
rsec
t are
the
solu
tions
of
the
equa
tion
f(x) =
g(x
); fin
d th
e ap
prox
imat
e so
lutio
ns u
sing
te
chno
logy
. ★
Incl
ude
case
s w
here
f(x)
and
/or g
(x) a
re li
near
, po
lyno
mia
l, ra
tiona
l, ab
solu
te v
alue
, exp
onen
tial,
and
loga
rithm
ic fu
nctio
ns.
Tas
ks m
ay in
volv
e an
y of
the
func
tion
type
s m
entio
ned
in th
e st
anda
rd.
65
where f(x) and/or g(x) are
linear, polynomial, rational,
absolute value, exponential, and logarithm
ic functions.★
A2.F.IF.A.3 3. R
ecognize that sequences are functions, som
etimes
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.
i) This standard is Supporting
work in A
lgebra II. This standard should support the M
ajor work in
F-BF.2 for coherence.
A2.F.IF.B.4 4. For a function that m
odels a relationship betw
een two
quantities, interpret key features of graphs and tables in term
s of the quantities, and sketch graphs show
ing key features given a verbal description of the relationship. Key features include: intercepts; intervals w
here the function is increasing, decreasing, positive, or negative; relative m
aximum
s and minim
ums;
symm
etries; end behavior; and periodicity.★
i) Tasks have a real-world context
ii) Tasks may involve polynom
ial, exponential, logarithm
ic, and trigonom
etric functions. Compare
note (ii) w
ith standard F-IF.7. The function types listed here are the sam
e as those listed in the Algebra II colum
n for standards F-IF.6 and F-IF.9.
A2.F.IF.A.1 For a function that m
odels a relationship between tw
o quantities, interpret key features of graphs and tables in term
s of the quantities, and sketch graphs show
ing key features given a verbal description of the relationship. ★
Key features include: intercepts; intervals w
here the function is increasing, decreasing, positive, or negative; relative m
aximum
s and minim
ums;
symm
etries; and end behavior. i) Tasks have a real-w
orld context. ii) Tasks m
ay involve square root, cube root, polynom
ial, exponential, and logarithmic functions.
A2.F.IF.B.6 6. C
alculate and interpret the average rate of change of a function (presented sym
bolically or as a table) over a specified interval. Estim
ate the rate of change from
a graph. ★
i) Tasks have a real-world context.
ii) Tasks may involve polynom
ial, exponential, logarithm
ic, and trigonom
etric functions. The function types listed here are the sam
e as those listed in the Algebra II colum
n for standards F-IF.4 and F-IF.9.
A2.F.IF.A.2 Calculate and interpret
the average rate of change of a function (presented sym
bolically or as a table) over a specified interval. Estim
ate the rate of change from a
graph.★
i) Tasks have a real-world context.
ii) Tasks may involve polynom
ial, exponential, and logarithm
ic function s.
A2.F.IF.C.7 7. G
raph functions expressed sym
bolically and show key
features of the graph, by hand
There is no additional scope or clarification inform
ation for this standard.
A2.F.IF.B.3 Graph functions
expressed symbolically and show
key features of the graph, by hand
A2.F.IF.B
.3a: Tasks are limited to square root and
cube root functions. The other functions are assessed in A
lgebra 1 .
66
in s
impl
e ca
ses
and
usin
g te
chno
logy
for m
ore
com
plic
ated
cas
es.★
c.G
raph
pol
ynom
ial f
unct
ions
,id
entif
ying
zer
os w
hen
suita
ble
fact
oriz
atio
ns a
re a
vaila
ble,
and
show
ing
end
beha
vior
.e.
Gra
ph e
xpon
entia
l and
loga
rithm
ic fu
nctio
ns, s
how
ing
i nte
rcep
ts a
nd e
nd b
ehav
ior,
and
trigo
nom
etric
func
tions
,sh
owin
g pe
riod,
mid
line,
and
ampl
itude
.
and
usin
g te
chno
logy
.★
a.G
raph
squ
are
root
, cub
e ro
ot,
and
piec
ewis
e de
fined
func
tions
,in
clud
ing
step
func
tions
and
abso
lute
val
ue fu
nctio
ns.
b.G
raph
pol
ynom
ial f
unct
ions
,id
entif
ying
zer
os w
hen
suita
ble
fact
oriz
atio
ns a
re a
vaila
ble
and
show
ing
end
beha
vior
.
c.G
raph
exp
onen
tial a
ndlo
garit
hmic
func
tions
, sho
win
gin
terc
epts
and
end
beh
avio
r.
A2.F
.IF.C
.8
8.W
rite
a fu
nctio
n de
fined
by
an e
xpre
ssio
n in
diff
eren
t but
equi
vale
nt fo
rms
to re
veal
and
expl
ain
diffe
rent
pro
perti
es o
fth
e fu
nctio
n.b.
Use
the
prop
ertie
s of
expo
nent
s to
inte
rpre
tex
pres
sion
s fo
r exp
onen
tial
func
tions
. For
exa
mpl
e,id
entif
y pe
rcen
t rat
e of
cha
nge
in fu
nctio
ns s
uch
as y
=(1
.02)
t , y
= (0
.97)
t ,y
= (1
.01)
12t ,
y =
(1.2
)t/10 ,
and
clas
sify
them
as
repr
esen
ting
expo
nent
ial g
row
th o
r dec
ay.
Ther
e is
no
addi
tiona
l sco
pe
or c
larif
icat
ion
info
rmat
ion
for
this
sta
ndar
d.
A2.F
.IF.B
.4 W
rite
a fu
nctio
n de
fined
by
an e
xpre
ssio
n in
di
ffere
nt b
ut e
quiv
alen
t for
ms
to
reve
al a
nd e
xpla
in d
iffer
ent
prop
ertie
s of
the
func
tion.
a.Kn
ow a
nd u
se th
e pr
oper
ties
ofex
pone
nts
to in
terp
ret e
xpre
ssio
nsf o
r exp
onen
tial f
unct
ions
.
For e
xam
ple,
iden
tify
perc
ent r
ate
of c
hang
e in
fu
nctio
ns s
uch
as y
= 2
x , y
= (1
/2)x ,
y =
2-x ,
y =
(1/2
)-x .
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is c
ours
e.
A2.F
.IF.C
.9
9.C
ompa
re p
rope
rties
of t
wo
func
tions
eac
h re
pres
ente
d in
a di
ffere
nt w
ay (a
lgeb
raic
ally
,gr
aphi
cally
, num
eric
ally
i nt a
bles
, or b
y ve
rbal
desc
riptio
ns).
For e
xam
ple,
give
n a
grap
h of
one
qua
drat
icfu
nctio
n an
d an
alg
ebra
icex
pres
sion
for a
noth
er, s
ayw
hich
has
the
larg
erm
axim
um.
i)Ta
sks
may
invo
lve
poly
nom
ial,
expo
nent
ial,
loga
rithm
ic, a
nd tr
igon
omet
ricfu
nctio
ns.
The
func
tion
type
s lis
ted
here
are
the
sam
e as
thos
e lis
ted
int h
e Al
gebr
a II
colu
mn
for
stan
dard
s F-
IF.4
and
F-IF
.6.
A2.F
.IF.B
.5 C
ompa
re p
rope
rties
of
two
func
tions
eac
h re
pres
ente
d in
a
diffe
rent
way
(alg
ebra
ical
ly,
grap
hica
lly, n
umer
ical
ly in
tabl
es, o
r by
ver
bal d
escr
iptio
ns).
Task
s m
ay in
volv
e po
lyno
mia
l, ex
pone
ntia
l, an
d lo
garit
hmic
func
tions
.
67
A2.F.BF.A.1 1. W
rite a function that describes a relationship betw
een two quantities.★
a. D
etermine an explicit
expression, a recursive process, or steps for calculation from
a context. b. C
ombine 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 m
odel.
For F-BF.1a: i) Tasks have a real-w
orld context ii) Tasks m
ay involve linear functions, quadratic functions, and exponential functions.
A2.F.BF.A.1 Write a function that
describes a relationship between
two quantities.★
a. D
etermine an explicit
expression, a recursive process, or steps for calculation from
a context. b. C
ombine standard function types
using arithmetic operations.
For example, given cost and revenue functions,
create a profit function. For A
2.F.BF.A
.1a: i) Tasks have a real-w
orld context. I i) Tasks m
ay involve linear functions, quadratic functions, and exponential functions.
A2.F.BF.A.2 2. W
rite arithmetic and
geometric sequences both
recursively and with an explicit
formula, use them
to model
situations, and translate betw
een the two form
s.★
There is no additional scope or clarification inform
ation for this standard.
A2.F.BF.A.2 Know and w
rite arithm
etic and geometric
sequences with an explicit form
ula and use them
to model situations.★
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A2.F.BF.B.3 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. Experim
ent 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
.
) Tasks may involve
polynomial, exponential,
logarithmic, and trigonom
etric functions ii) Tasks m
ay involve recognizing even and odd functions. The function types listed in note (i) are the sam
e as those listed in the Algebra II colum
n for standards F-IF.4, F-IF.6, and F-IF.9.
A2.F.BF.B.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. Experim
ent with
cases and illustrate an explanation of the effects on the graph using technology.
i) Tasks may involve polynom
ial, exponential, and logarithm
ic functions. ii) Tasks m
ay involve recognizing even and odd functions.
A2.F.BF.B.4 4. Find inverse functions. a. Solve an equation of the form
f(x) = c for a simple
function f that has an inverse
There is no additional scope or clarification inform
ation for this standard.
A2.F.BF.B.4 Find inverse functions. a. Find the inverse of a function
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
68
and
writ
e an
exp
ress
ion
for t
he
inve
rse.
For
exa
mpl
e, f(
x) =
2x3
or f(
x) =
(x+1
)/(x–
1) fo
r x =
1.
whe
n th
e gi
ven
func
tion
is o
ne-to
-on
e.
A2.F
.LE.
A.2
2. C
onst
ruct
line
ar a
nd
expo
nent
ial f
unct
ions
, in
clud
ing
arith
met
ic a
nd
geom
etric
seq
uenc
es, g
iven
a
grap
h, a
des
crip
tion
of a
re
latio
nshi
p, o
r tw
o in
put-
outp
ut p
airs
(inc
lude
read
ing
thes
e fro
m a
tabl
e).
i) Ta
sks
will
incl
ude
solv
ing
mul
ti-st
ep p
robl
ems
by
cons
truct
ing
linea
r and
ex
pone
ntia
l fun
ctio
ns.
A2.F
.LE.
A.1
Con
stru
ct li
near
and
ex
pone
ntia
l fun
ctio
ns, i
nclu
ding
ar
ithm
etic
and
geo
met
ric
sequ
ence
s, g
iven
a g
raph
, a ta
ble,
a
desc
riptio
n of
a re
latio
nshi
p, o
r in
put-o
utpu
t pai
rs.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is c
ours
e.
A2.F
.LE.
A.4
4. F
or e
xpon
entia
l mod
els,
ex
pres
s as
a lo
garit
hm th
e so
lutio
n to
abct
= d
whe
re a
, c,
and
d ar
e nu
mbe
rs a
nd th
e ba
se b
is 2
, 10,
or e
; eva
luat
e th
e lo
garit
hm u
sing
te
chno
logy
.
Ther
e is
no
addi
tiona
l sco
pe
or c
larif
icat
ion
info
rmat
ion
for
this
sta
ndar
d
A2.F
.LE.
A.2
For e
xpon
entia
l m
odel
s, e
xpre
ss a
s a
loga
rithm
the
solu
tion
to a
bc t =
d w
here
a, c
, and
d
are
num
bers
and
the
base
b is
2,
10, o
r e; e
valu
ate
the
loga
rithm
us
ing
tech
nolo
gy.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is c
ours
e.
A2.F
.LE.
B.5
5. In
terp
ret t
he p
aram
eter
s in
a
linea
r or e
xpon
entia
l fun
ctio
n in
term
s of
a c
onte
xt.
i) Ta
sks
have
a re
al-w
orld
co
ntex
t. ii)
Tas
ks a
re lim
ited
to
expo
nent
ial f
unct
ions
with
do
mai
ns n
ot in
the
inte
gers
.
A2.F
.LE.
B.3
Inte
rpre
t the
pa
ram
eter
s in
a li
near
or
expo
nent
ial f
unct
ion
in te
rms
of a
co
ntex
t.
For e
xam
ple,
the
equa
tion
y =
5000
(1.0
6)x m
odel
s th
e ris
ing
popu
latio
n of
a c
ity w
ith 5
000
resi
dent
s w
hen
the
annu
al g
row
th ra
te is
6 p
erce
nt. W
hat w
ill be
the
effe
ct o
n th
e eq
uatio
n if
the
city
's g
row
th ra
te
was
7 p
erce
nt in
stea
d of
6 p
erce
nt?
Th
ere
are
no a
sses
smen
t lim
its fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
cou
rse.
A2.F
.TF.
A.1
1. U
nder
stan
d ra
dian
mea
sure
of
an
angl
e as
the
leng
th o
f th
e ar
c on
the
unit
circ
le
subt
ende
d by
the
angl
e x.
Ther
e is
no
addi
tiona
l sco
pe
or c
larif
icat
ion
info
rmat
ion
for
this
sta
ndar
d.
A2.F
.TF.
A.1
Und
erst
and
and
use
radi
an m
easu
re o
f an
angl
e.
a. U
nder
stan
d ra
dian
mea
sure
of
an a
ngle
as
the
leng
th o
f the
arc
on
the
unit
circ
le s
ubte
nded
by
the
angl
e.
b. U
se th
e un
it ci
rcle
to fi
nd s
in θ
, co
s θ,
and
tan
θ w
hen
θ is
a
com
mon
ly re
cogn
ized
ang
le
betw
een
0 an
d 2π
.
Com
mon
ly re
cogn
ized
ang
les
incl
ude
all m
ultip
les
nπ /6
and
nπ
/4, w
here
n is
an
inte
ger.
Ther
e ar
e no
as
sess
men
t lim
its fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
cou
rse.
A2.F
.TF.
A.2
2. E
xpla
in h
ow th
e un
it ci
rcle
Th
ere
is n
o ad
ditio
nal s
cope
A2
.F.T
F.A.
2 Ex
plai
n ho
w th
e un
it Th
ere
are
no a
sses
smen
t lim
its fo
r thi
s st
anda
rd.
69
in the coordinate plane enables the extension of trigonom
etric functions to all real num
bers, interpreted as radian m
easures of angles traversed counterclockw
ise around the unit circle.
or clarification information for
this standard. circle in the coordinate plane enables the extension of trigonom
etric functions to all real num
bers, interpreted as radian m
easures of angles traversed counterclockw
ise around the unit circle.
The entire standard is assessed in this course.
A2.F.TF.B.5 5. C
hoose trigonometric
functions to model periodic
phenomena w
ith specified am
plitude, frequency, and m
idline.★
There is no additional scope or clarification inform
ation for this standard.
A2.F.TF.C.8 8. Prove the Pythagorean identity sin
2(θ) + cos2(θ) = 1
and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
There is no additional scope or clarification inform
ation for this standard.
A2.F.TF.B.3 Know
and use trigonom
etric identities to find values of trig functions. a. G
iven a point on a circle centered at the origin, recognize and use the right triangle ratio definitions of sin θ, cos θ, and tan θ to evaluate the trigonom
etric functions. b. G
iven the quadrant of the angle, use the identity sin
2 θ + cos2 θ = 1
to find sin θ given cos θ, or vice versa.
Com
monly recognized angles include all m
ultiples nπ
/6 and nπ /4, w
here n is an integer. There are no assessm
ent limits for this standard.
The entire standard is assessed in this course.
A2.GPE.A.2
2. Derive the equation of a
parabola given a focus and directrix.
There is no additional scope or clarification inform
ation for this standard.
A2.S.ID.A.4 4. U
se the mean and standard
There is no additional scope A2.S.ID.A.1 U
se the mean and
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
70
devi
atio
n of
a d
ata
set t
o fit
it
to a
nor
mal
dis
tribu
tion
and
to
estim
ate
popu
latio
n pe
rcen
tage
s. R
ecog
nize
that
th
ere
are
data
set
s fo
r whi
ch
such
a p
roce
dure
is n
ot
appr
opria
te. U
se c
alcu
lato
rs,
spre
adsh
eets
, and
tabl
es to
es
timat
e ar
eas
unde
r the
no
rmal
cur
ve.
or c
larif
icat
ion
info
rmat
ion
for
this
sta
ndar
d.
stan
dard
dev
iatio
n of
a d
ata
set t
o fit
it to
a n
orm
al d
istri
butio
n an
d to
es
timat
e po
pula
tion
perc
enta
ges
usin
g th
e Em
piric
al R
ule.
A2.S
.ID.B
.6
6.R
epre
sent
dat
a on
two
quan
titat
ive
varia
bles
on
asc
atte
r plo
t, an
d de
scrib
e ho
wth
e va
riabl
es a
re re
late
d.a.
Fit a
func
tion
to th
e da
ta;
use
func
tions
fitte
d to
dat
a to
s olv
e pr
oble
ms
in th
e co
ntex
tof
the
data
. Use
giv
enfu
nctio
ns o
r cho
ose
a fu
nctio
ns u
gges
ted
by th
e co
ntex
t.Em
phas
ize
linea
r, qu
adra
tic,
and
expo
nent
ial m
odel
s.
i)Ta
sks
have
a re
al-w
orld
cont
ext.
ii)Ta
sks
are
limite
d to
expo
nent
ial f
unct
ions
wit h
dom
ains
not
in th
e in
tege
rsan
d tri
gono
met
ricfu
nctio
ns.
A2.S
.ID.B
.2 R
epre
sent
dat
a on
two
quan
titat
ive
varia
bles
on
a sc
atte
r pl
ot, a
nd d
escr
ibe
how
the
varia
bles
are
rela
ted.
a.Fi
t a fu
nctio
n to
the
data
; use
func
tions
fitte
d to
dat
a to
sol
vepr
oble
ms
in th
e co
ntex
t of t
he d
ata.
Use
giv
en fu
nctio
ns o
r cho
ose
a fu
nctio
n su
gges
ted
by th
e co
ntex
t.
Em
phas
ize
linea
r, qu
adra
tic, a
nd e
xpon
entia
l m
odel
s.
i)Ta
sks
have
a re
al-w
orld
con
text
.ii)
Task
s ar
e lim
ited
to e
xpon
entia
l fun
ctio
ns w
ithdo
mai
ns n
ot in
the
inte
gers
.
A2.S
.IC.A
.1
1.U
nder
stan
d st
atis
tics
as a
proc
ess
for m
akin
g in
fere
nces
abou
t pop
ulat
ion
para
met
ers
base
d on
a ra
ndom
sam
ple
from
that
pop
ulat
ion.
Ther
e is
no
addi
tiona
l sco
pe
or c
larif
icat
ion
info
rmat
ion
for
this
sta
ndar
d.
A2.S
.IC.A
.2
2.D
ecid
e if
a sp
ecifi
ed m
odel
is c
onsi
sten
t with
resu
lts fr
oma
give
n da
ta-g
ener
atin
gpr
oces
s, e
.g.,
usi n
gs i
mul
atio
n. F
or e
xam
ple,
am
odel
say
s a
spin
ning
coi
nfa
lls h
eads
up
with
pro
babi
lity
0.5.
Wou
ld a
resu
lt of
5 ta
ils in
a ro
w c
ause
you
to q
uest
ion
the
mod
el?
Ther
e is
no
addi
tiona
l sco
pe
or c
larif
icat
ion
info
rmat
ion
for
this
sta
ndar
d.
A2.S
.IC.B
.3
3.R
ecog
nize
the
purp
oses
of
and
diffe
renc
es a
mon
g sa
mpl
eTh
ere
is n
o ad
ditio
nal s
cope
or
cla
rific
atio
n in
form
atio
n fo
r A2
.S.IC
.A.1
Rec
ogni
ze th
e pu
rpos
es o
f and
diff
eren
ces
amon
g Fo
r exa
mpl
e, in
a g
iven
situ
atio
n, is
it m
ore
appr
opria
te to
use
a s
ampl
e su
rvey
, an
expe
rimen
t,
71
surveys, experiments, and
observational studies; explain how
randomization relates to
each.
this standard. sam
ple surveys, experiments, and
observational studies; explain how
randomization relates to each.
or an observational study? Explain how
random
ization affects the bias in a study. There are no assessm
ent limits for this standard.
The entire standard is assessed in this course.
A2.S.IC.B.4 4. U
se data from a sam
ple survey to estim
ate a population m
ean or proportion; develop a m
argin of error through the use of sim
ulation m
odels for random sam
pling.
There is no additional scope or clarification inform
ation for this standard.
A2.S.IC.A.2 Use data from
a sam
ple survey to estimate a
population mean or proportion; use
a given margin of error to solve a
problem in context
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A2.S.IC.B.5 5. U
se data from a
randomized experim
ent to com
pare two treatm
ents; use sim
ulations to decide if differences betw
een param
eters are significant.
There is no additional scope or clarification inform
ation for this standard.
A2.S.IC.B.6 6. Evaluate reports based on data.
There is no additional scope or clarification inform
ation for this standard.
A2.S.CP.A.1 1. D
escribe events as subsets of a sam
ple space (the set of outcom
es) using characteristics (or categories) of the outcom
es, or as unions, intersections, or com
plements
of other events (“or,” “and,” “not”).
There is no additional scope or clarification inform
ation for this standard.
A2.S.CP.A.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 com
plements of other events (“or,”
“and,” “not”).
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A2.S.CP.A.2 2. U
nderstand 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 determ
ine if they are independent.
There is no additional scope or clarification inform
ation for this standard.
A2.S.CP.A.2 Understand that tw
o 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 determ
ine if they are independent.
There are no assessment lim
its for this standard. The entire standard is assessed in this course
A2.S.CP.A.3 3. U
nderstand the conditional There is no additional scope
A2.S.CP.A.3 Know and understand
There are no assessment lim
its for this standard.
72
prob
abilit
y of
A g
iven
B a
s P(
A an
d B)
/P(B
), an
d in
terp
ret
inde
pend
ence
of A
and
B a
s sa
ying
that
the
cond
ition
al
prob
abilit
y of
A g
iven
B is
the
sam
e as
the
prob
abilit
y of
A,
and
the
cond
ition
al p
roba
bilit
y of
B g
iven
A is
the
sam
e as
th
e pr
obab
ility
of B
.
or c
larif
icat
ion
info
rmat
ion
for
this
sta
ndar
d.
the
cond
ition
al p
roba
bilit
y of
A
give
n B
as P
(A a
nd B
)/P(B
), an
d in
terp
ret i
ndep
ende
nce
of A
and
B
as s
ayin
g th
at th
e co
nditi
onal
pr
obab
ility
of A
giv
en B
is th
e sa
me
as th
e pr
obab
ility
of A
, and
the
cond
ition
al p
roba
bilit
y of
B g
iven
A
is th
e sa
me
as th
e pr
obab
ility
of B
.
The
entir
e st
anda
rd is
ass
esse
d in
this
cou
rse
A2.S
.CP.
A.4
4.C
onst
ruct
and
inte
rpre
t tw
o-w
ay fr
eque
ncy
tabl
es o
f dat
aw
hen
two
cate
gorie
s ar
eas
soci
ated
with
eac
h ob
ject
bein
g cl
assi
fied.
Use
the
two-
way
tabl
e as
a s
ampl
e sp
ace
to d
ecid
e if
even
ts a
rein
depe
nden
t and
toap
prox
imat
e co
nditi
onal
prob
abilit
ies.
For
exa
mpl
e,co
llect
dat
a fro
m a
rand
omsa
mpl
e of
stu
dent
s in
you
rsc
hool
on
thei
r fav
orite
sub
ject
amon
g m
ath,
sci
ence
, and
Engl
ish.
Est
imat
e th
epr
obab
ility
that
a ra
ndom
lyse
lect
ed s
tude
nt fr
om y
our
scho
ol w
ill fa
vor s
cien
ce g
iven
that
the
stud
ent i
s in
tent
hgr
ade.
Do
the
sam
e fo
r oth
ersu
bjec
ts a
nd c
ompa
re th
ere
sults
.
Ther
e is
no
addi
tiona
l sco
pe
or c
larif
icat
ion
info
rmat
ion
for
this
sta
ndar
d.
A2.S
.CP.
A.5
5.R
ecog
nize
and
exp
lain
the
conc
epts
of c
ondi
tiona
lpr
obab
ility
and
inde
pend
ence
in e
very
day
lang
uage
and
e ver
yday
situ
atio
ns. F
orex
ampl
e, c
ompa
re th
e ch
ance
of h
avin
g lu
ng c
ance
r if y
ouar
e a
smok
er w
ith th
e ch
anc e
of b
eing
a s
mok
er if
you
hav
elu
ng c
ance
r.
Ther
e is
no
addi
tiona
l sc
ope
or c
larif
icat
ion
info
rmat
ion
for t
his
stan
dard
.
A2.S
.CP.
A.4
Rec
ogni
ze a
nd
expl
ain
the
conc
epts
of c
ondi
tiona
l pr
obab
ility
and
inde
pend
ence
in
ever
yday
lang
uage
and
eve
ryda
y si
tuat
ions
.
For e
xam
ple,
com
pare
the
chan
ce o
f hav
ing
lung
ca
ncer
if y
ou a
re a
sm
oker
with
the
chan
ce o
f bei
ng
a sm
oker
if y
ou h
ave
lung
can
cer.
The
re a
re n
o as
sess
men
t lim
its fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
cou
rse.
73
A2.S.CP.B.6 6. Find the conditional probability of A given B as the fraction of B’s outcom
es that also belong to A, and interpret the answ
er in terms of the
model.
There is no additional scope or clarification inform
ation for this standard.
A2.S.CP.B.5 Find the conditional probability of A given B as the fraction of B’s outcom
es that also belong to A and interpret the answ
er in terms of the m
odel.
For example, a teacher gave tw
o exams. 75 percent
passed the first quiz and 25 percent passed both. W
hat percent who passed the first quiz also passed
the second quiz? There are no assessm
ent limits for this standard.
The entire standard is assessed in this course.
A2.S.CP.B.7 7. Apply the Addition R
ule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answ
er in terms of the m
odel.
There is no additional scope or clarification inform
ation for this standard.
A2.S.CP.B.6 Know and apply the
Addition Rule, P(A or B) = P(A) +
P(B) – P(A and B), and interpret the answ
er in terms of the m
odel.
For example, in a m
ath class of 32 students, 14 are boys and 18 are girls. O
n a unit test 6 boys and 5 girls m
ade an A. If a student is chosen at random
from
a class, what is the probability of choosing a girl
or an A student?
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
M
ajor Work of the G
rade
Supporting Work
74
Not
atio
n TN
Sta
ndar
ds th
roug
h M
ay 2
017
Revi
sed
TN S
tand
ards
St
anda
rd
C
lari
ficat
ion
G.CO
.A.1
Kn
ow p
reci
se d
efin
ition
s of a
ngle
, circ
le,
perp
endi
cula
r lin
e, p
aral
lel l
ine,
and
line
se
gmen
t, ba
sed
on th
e un
defin
ed n
otio
ns
of p
oint
, lin
e, d
istan
ce a
long
a li
ne, a
nd
dist
ance
aro
und
a ci
rcul
ar a
rc.
G.CO
.A.1
Kno
w p
reci
se d
efin
ition
s of a
ngle
, ci
rcle
, per
pend
icul
ar li
ne, p
aral
lel l
ine,
and
lin
e se
gmen
t, ba
sed
on th
e un
defin
ed
notio
ns o
f poi
nt, l
ine,
pla
ne, d
istan
ce a
long
a
line,
and
dist
ance
aro
und
a ci
rcul
ar a
rc.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.CO
.A.2
R
epre
sent
tran
sfor
mat
ions
in th
e pl
ane
usin
g, e
.g.,
tran
spar
enci
es a
nd g
eom
etry
so
ftwar
e; d
escr
ibe
tran
sfor
mat
ions
as
func
tions
that
take
poi
nts i
n th
e pl
ane
as
inpu
ts a
nd g
ive
othe
r poi
nts a
s out
puts
. Co
mpa
re tr
ansf
orm
atio
ns th
at p
rese
rve
dist
ance
and
ang
le to
thos
e th
at d
o no
t (e
.g.,
tran
slatio
n ve
rsus
hor
izont
al
stre
tch)
.
G.CO
.A.2
Rep
rese
nt tr
ansf
orm
atio
ns in
the
plan
e in
mul
tiple
way
s, in
clud
ing
tech
nolo
gy.
Desc
ribe
tran
sfor
mat
ions
as f
unct
ions
that
ta
ke p
oint
s in
the
plan
e (p
re-im
age)
as i
nput
s an
d gi
ve o
ther
poi
nts (
imag
e) a
s out
puts
. Co
mpa
re tr
ansf
orm
atio
ns th
at p
rese
rve
dist
ance
and
ang
le m
easu
re to
thos
e th
at d
o no
t (e.
g., t
rans
latio
n ve
rsus
hor
izont
al
stre
tch)
.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.CO
.A.3
G
iven
a re
ctan
gle,
par
alle
logr
am,
trap
ezoi
d, o
r reg
ular
pol
ygon
, des
crib
e th
e ro
tatio
ns a
nd re
flect
ions
that
car
ry it
ont
o its
elf.
G.CO
.A.3
Giv
en a
rect
angl
e, p
aral
lelo
gram
, tr
apez
oid,
or r
egul
ar p
olyg
on, d
escr
ibe
the
rota
tions
and
refle
ctio
ns th
at c
arry
the
shap
e on
to it
self.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.CO
.A.4
D
evel
op d
efin
ition
s of r
otat
ions
, re
flect
ions
, and
tran
slatio
ns in
term
s of
angl
es, c
ircle
s, pe
rpen
dicu
lar l
ines
, par
alle
l lin
es, a
nd li
ne se
gmen
ts.
G.CO
.A.4
Dev
elop
def
initi
ons o
f rot
atio
ns,
refle
ctio
ns, a
nd tr
ansla
tions
in te
rms o
f an
gles
, circ
les,
perp
endi
cula
r lin
es, p
aral
lel
lines
, and
line
segm
ents
.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.CO
.A.5
G
iven
a g
eom
etric
figu
re a
nd a
rota
tion,
re
flect
ion,
or t
rans
latio
n, d
raw
the
tran
sfor
med
figu
re u
sing,
e.g
., gr
aph
pape
r, tr
acin
g pa
per,
or g
eom
etry
so
ftwar
e. S
peci
fy a
sequ
ence
of
tran
sfor
mat
ions
that
will
carr
y a
give
n fig
ure
onto
ano
ther
.
G.C
O.A
.5 G
iven
a g
eom
etric
figu
re a
nd a
rig
id m
otio
n, d
raw
the
imag
e of
the
figur
e in
m
ultip
le w
ays,
incl
udin
g te
chno
logy
. Spe
cify
a
sequ
ence
of r
igid
mot
ions
that
will
carr
y a
give
n fig
ure
onto
ano
ther
.
Rigi
d m
otio
ns in
clud
e ro
tatio
ns, r
efle
ctio
ns,
and
tran
slatio
ns.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
75
G.CO.B.6
Use geom
etric descriptions of rigid m
otions 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.
G.CO.B.6 U
se geometric descriptions of rigid
motions to transform
figures and to predict the effect of a given rigid m
otion on a given figure; given tw
o figures, use the definition of congruence in term
s of rigid motions to
determine inform
ally if they are congruent.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G.CO.B.7
Use the definition of congruence in term
s of rigid m
otions to show that tw
o triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.B.7 U
se the definition of congruence in term
s 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.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G.CO.B.8
Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow
from the definition of congruence in term
s of rigid m
otions.
G.CO.B.8 Explain how
the criteria for triangle congruence (ASA, SAS, AAS, and SSS) follow
from
the definition of congruence in terms of
rigid motions.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G.CO.C.9
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 segm
ent are exactly those equidistant from
the segment’s
endpoints.
G.CO.C.9 Prove theorem
s about lines and angles.
Proving includes, but is not limited to,
completing partial proofs; constructing tw
o-colum
n or paragraph proofs; using transform
ations to prove theorems; analyzing
proofs; and critiquing completed proofs.
Theorems include but are not lim
ited to: vertical angles are congruent; w
hen a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segm
ent are exactly those equidistant from
the segm
ent’s endpoints.
76
G.CO
.C.1
0 P
rove
theo
rem
s abo
ut tr
iang
les.
Th
eore
ms i
nclu
de: m
easu
res o
f int
erio
r an
gles
of a
tria
ngle
sum
to 1
80°;
base
an
gles
of i
sosc
eles
tria
ngle
s are
co
ngru
ent;
the
segm
ent j
oini
ng m
idpo
ints
of
two
sides
of a
tria
ngle
is p
aral
lel t
o th
e th
ird si
de a
nd h
alf t
he le
ngth
; the
med
ians
of
a tr
iang
le m
eet a
t a p
oint
.
G.CO
.C.1
0 Pr
ove
theo
rem
s abo
ut tr
iang
les.
Prov
ing
incl
udes
, but
is n
ot li
mite
d to
, co
mpl
etin
g pa
rtia
l pro
ofs;
cons
truc
ting
two-
colu
mn
or p
arag
raph
pro
ofs;
usin
g tr
ansf
orm
atio
ns to
pro
ve th
eore
ms;
ana
lyzin
g pr
oofs
; and
criti
quin
g co
mpl
eted
pro
ofs.
Th
eore
ms i
nclu
de b
ut a
re n
ot li
mite
d to
: m
easu
res o
f int
erio
r ang
les o
f a tr
iang
le su
m
to 1
80°;
base
ang
les o
f iso
scel
es tr
iang
les a
re
cong
ruen
t; th
e se
gmen
t joi
ning
mid
poin
ts o
f tw
o sid
es o
f a tr
iang
le is
par
alle
l to
the
third
sid
e an
d ha
lf th
e le
ngth
; the
med
ians
of a
tr
iang
le m
eet a
t a p
oint
G.CO
.C.1
1
Pro
ve th
eore
ms a
bout
par
alle
logr
ams.
Theo
rem
s inc
lude
: opp
osite
side
s are
co
ngru
ent,
oppo
site
angl
es a
re co
ngru
ent,
the
diag
onal
s of a
par
alle
logr
am b
isect
ea
ch o
ther
, and
conv
erse
ly, r
ecta
ngle
s are
pa
ralle
logr
ams w
ith co
ngru
ent d
iago
nals.
G.CO
.C.1
1 P
rove
theo
rem
s abo
ut
para
llelo
gram
s.
Prov
ing
incl
udes
, but
is n
ot li
mite
d to
, co
mpl
etin
g pa
rtia
l pro
ofs;
cons
truc
ting
two-
colu
mn
or p
arag
raph
pro
ofs;
usin
g tr
ansf
orm
atio
ns to
pro
ve th
eore
ms;
ana
lyzin
g pr
oofs
; and
criti
quin
g co
mpl
eted
pro
ofs.
Th
eore
ms i
nclu
de b
ut a
re n
ot li
mite
d to
: op
posit
e sid
es a
re co
ngru
ent,
oppo
site
angl
es
are
cong
ruen
t, th
e di
agon
als o
f a
para
llelo
gram
bise
ct e
ach
othe
r, an
d co
nver
sely
, rec
tang
les a
re p
aral
lelo
gram
s w
ith co
ngru
ent d
iago
nals.
77
G.CO.D.12
Make form
al geometric constructions
with a variety of tools and m
ethods (com
pass and straightedge, string, reflective devices, paper folding, dynam
ic geom
etric software, etc.). Copying a
segment; copying an angle; bisecting a
segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segm
ent; and constructing a line parallel to a given line through a point not on the line.
G.CO.D.12 M
ake formal geom
etric constructions w
ith a variety of tools and m
ethods (compass and straightedge, string,
reflective devices, paper folding, dynamic
geometric softw
are, etc.).
Constructions include but are not limited to:
copying a segment; copying an angle;
bisecting a segment; bisecting an angle;
constructing perpendicular lines, including the perpendicular bisector of a line segm
ent; constructing a line parallel to a given line through a point not on the line, and constructing the follow
ing objects inscribed in a circle: an equilateral triangle, square, and a regular hexagon.
G.CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
G.SRT.A.1 Verify experim
entally 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 segm
ent is longer or shorter in the ratio given by the scale factor.
G.SRT.A.1 Verify informally the properties of
dilations given by a center and a scale factor.
Properties include but are not limited to: 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 of the dilation unchanged; the dilation of a line segm
ent is longer or shorter in the ratio given by the scale factor.
G.SRT.A.2 Given tw
o figures, use the definition of sim
ilarity in terms of sim
ilarity transform
ations to decide if they are sim
ilar; explain using similarity
transformations the m
eaning of similarity
for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G.SRT.A.2 Given two figures, use the
definition of similarity in term
s of similarity
transformations to decide if they are sim
ilar; explain using sim
ilarity transformations the
meaning of sim
ilarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
78
G.SR
T.A.
3U
se th
e pr
oper
ties o
f sim
ilarit
y tr
ansf
orm
atio
ns to
est
ablis
h th
e AA
cr
iterio
n fo
r tw
o tr
iang
les t
o be
sim
ilar.
G.SR
T.A.
3 U
se th
e pr
oper
ties o
f sim
ilarit
ytr
ansf
orm
atio
ns to
est
ablis
h th
e AA
crite
rion
for t
wo
tria
ngle
s to
be si
mila
r.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.SR
T.B.
4 P
rove
theo
rem
s abo
ut tr
iang
les.
Th
eore
ms i
nclu
de: a
line
par
alle
l to
one
side
of a
tria
ngle
div
ides
the
othe
r tw
o pr
opor
tiona
lly, a
nd co
nver
sely
; the
Py
thag
orea
n Th
eore
m p
rove
d us
ing
tria
ngle
sim
ilarit
y.
G.SR
T.B.
4 Pr
ove
theo
rem
s abo
ut si
mila
rtr
iang
les.
Prov
ing
incl
udes
, but
is n
ot li
mite
d to
, co
mpl
etin
g pa
rtia
l pro
ofs;
cons
truc
ting
two-
colu
mn
or p
arag
raph
pro
ofs;
usin
g tr
ansf
orm
atio
ns to
pro
ve th
eore
ms;
ana
lyzin
g pr
oofs
; and
criti
quin
g co
mpl
eted
pro
ofs.
Theo
rem
s inc
lude
but
are
not
lim
ited
to: a
lin
e pa
ralle
l to
one
side
of a
tria
ngle
div
ides
th
e ot
her t
wo
prop
ortio
nally
, and
conv
erse
ly;
the
Pyth
agor
ean
Theo
rem
pro
ved
usin
g tr
iang
le si
mila
rity.
G.SR
T.B.
5 U
se co
ngru
ence
and
sim
ilarit
y cr
iteria
for
tria
ngle
s to
solv
e pr
oble
ms a
nd to
pro
ve
rela
tions
hips
in g
eom
etric
figu
res.
G.SR
T.B.
5 U
se co
ngru
ence
and
sim
ilarit
ycr
iteria
for t
riang
les t
o so
lve
prob
lem
s and
toju
stify
rela
tions
hips
in g
eom
etric
figu
res.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.SR
T.C.
6 U
nder
stan
d th
at b
y sim
ilarit
y, si
de ra
tios
in ri
ght t
riang
les a
re p
rope
rtie
s of t
he
angl
es in
the
tria
ngle
, lea
ding
to
defin
ition
s of t
rigon
omet
ric ra
tios f
or
acut
e an
gles
.
G.SR
T.C.
6 U
nder
stan
d th
at b
y sim
ilarit
y, si
dera
tios i
n rig
ht tr
iang
les a
re p
rope
rtie
s of t
hean
gles
in th
e tr
iang
le, l
eadi
ng to
def
initi
ons
of tr
igon
omet
ric ra
tios f
or a
cute
ang
les.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.SR
T.C.
7 E
xpla
in a
nd u
se th
e re
latio
nshi
p be
twee
n th
e sin
e an
d co
sine
of c
ompl
emen
tary
an
gles
.
G.SR
T.C.
7 Ex
plai
n an
d us
e th
e re
latio
nshi
pbe
twee
n th
e sin
e an
d co
sine
ofco
mpl
emen
tary
ang
les.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
79
G.SRT.C.8 U
se trigonometric ratios and the
Pythagorean Theorem to solve right
triangles in applied problems. ★
G.SRT.C.8 Solve triangles. ★
a. Know
and use trigonometric ratios
and the Pythagorean Theorem to
solve right triangles in applied problem
s.
b. Know
and use the Law of Sines and
Law of Cosines to solve problem
s in real life situations. Recognize w
hen it is appropriate to use each.
Ambiguous cases w
ill not be included in assessm
ent.
G.C.A.1 Prove that all circles are sim
ilar. G.C.A.1 Recognize that all circles are sim
ilar.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G.C.A.2 Identify and describe relationships am
ong inscribed angles, radii, and chords. Include the relationship betw
een central, inscribed, and circum
scribed angles; inscribed angles on a diam
eter are right angles; the radius of a circle is perpendicular to the tangent w
here the radius intersects the circle.
G.C.A.2 Identify and describe relationships am
ong 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 w
here the radius intersects the circle, and properties of angles for a quadrilateral inscribed in a circle.
G.C.A.3 Construct the inscribed and circum
scribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
G.C.A.5 Derive using sim
ilarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian m
easure of the angle as the constant of proportionality; derive the form
ula for the area of a sector.
80
G.C.
A.3
Con
stru
ct th
e in
cent
er a
ndci
rcum
cent
er o
f a tr
iang
le a
nd u
se th
eir
prop
ertie
s to
solv
e pr
oble
ms i
n co
ntex
t.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.C.
B.4
Know
the
form
ula
and
find
the
area
of a
sect
or o
f a ci
rcle
in a
real
-wor
ld co
ntex
t.
For e
xam
ple,
use
pro
port
iona
l rel
atio
nshi
ps
and
angl
es m
easu
red
in d
egre
es o
r rad
ians
.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.GP
E.A.
1 D
eriv
e th
e eq
uatio
n of
a c
ircle
of g
iven
ce
nter
and
radi
us u
sing
the
Pyth
agor
ean
Theo
rem
; com
plet
e th
e sq
uare
to fi
nd th
e ce
nter
and
radi
us o
f a ci
rcle
giv
en b
y an
eq
uatio
n.
G.GP
E.A.
1 Kn
ow a
nd w
rite
the
equa
tion
of a
circ
le o
f giv
en ce
nter
and
radi
us u
sing
the
Pyth
agor
ean
Theo
rem
.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
G.GP
E.B.
4 U
se co
ordi
nate
s to
prov
e sim
ple
geom
etric
theo
rem
s alg
ebra
ical
ly. F
or
exam
ple,
pro
ve o
r disp
rove
that
a fi
gure
de
fined
by
four
giv
en p
oint
s in
the
coor
dina
te p
lane
is a
rect
angl
e; p
rove
or
disp
rove
that
the
poin
t (1,
√3,
) lie
s on
the
circ
le ce
nter
ed a
t the
orig
in a
nd
cont
aini
ng th
e po
int (
0, 2
).
G.GP
E.B.
2 U
se c
oord
inat
es to
pro
ve si
mpl
ege
omet
ric th
eore
ms a
lgeb
raic
ally
.
For e
xam
ple,
pro
ve o
r disp
rove
that
a fi
gure
de
fined
by
four
giv
en p
oint
s in
the
coor
dina
te
plan
e is
a re
ctan
gle;
pro
ve o
r disp
rove
that
th
e po
int (
1, √
3) li
es o
n th
e ci
rcle
cent
ered
at
the
orig
in a
nd co
ntai
ning
the
poin
t (0,
2).
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
. G.
GPE.
B.5
Pro
ve th
e slo
pe cr
iteria
for p
aral
lel a
nd
perp
endi
cula
r lin
es a
nd u
se th
em to
solv
e ge
omet
ric p
robl
ems (
e.g.
, fin
d th
e eq
uatio
n of
a li
ne p
aral
lel o
r pe
rpen
dicu
lar t
o a
give
n lin
e th
at p
asse
s th
roug
h a
give
n po
int).
G.GP
E.B.
3 Pr
ove
the
slope
crit
eria
for p
aral
lel
and
perp
endi
cula
r lin
es a
nd u
se th
em to
solv
e ge
omet
ric p
robl
ems.
For e
xam
ple,
find
the
equa
tion
of a
line
pa
ralle
l or p
erpe
ndic
ular
to a
giv
en li
ne th
at
pass
es th
roug
h a
give
n po
int.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
81
G.GPE.B.6 6. Find the point on a directed line segm
ent between tw
o given points that partitions the segm
ent in a given ratio.
G.GPE.B.4 Find the point on a directed line segm
ent between tw
o given points that partitions the segm
ent in a given ratio.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G.GPE.B.7 7. U
se coordinates to compute perim
eters of polygons and areas of triangles and rectangles, e.g., using the distance form
ula. ★
G.GPE.B.5 Know and use coordinates to
compute perim
eters of polygons and areas of triangles and rectangles.★
For example, use the distance form
ula. There are no assessm
ent limits for this standard.
The entire standard is assessed in this course.
G.GMD.A.1
1. Give an informal argum
ent for the form
ulas for the circumference of a circle,
area of a circle, volume of a cylinder,
pyramid, and cone. U
se dissection argum
ents, Cavalieri’s principle, and inform
al limit argum
ents.
G.GMD.A.1 Give an inform
al argument for
the formulas for the circum
ference of a circle and the volum
e and surface area of a cylinder, cone, prism
, and pyramid.
Informal argum
ents may include but are not
limited to using the dissection argum
ent, applying Cavalieri’s principle, and constructing inform
al limit argum
ents. There are no assessm
ent limits for this
standard. The entire standard is assessed in this course
G.GMD.A.3
3. Use volum
e formulas for cylinders,
pyramids, cones, and spheres to solve
problems. ★
G.GMD.A.2 Know
and use volume and
surface area formulas for cylinders, cones,
prisms, pyram
ids, and spheres to solve problem
s.★
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G.GMD.A.4
4. Identify the shapes of two-dim
ensional cross-sections of three dim
ensional objects, and identify three-dim
ensional objects generated by rotations of tw
o-dim
ensional objects.
82
G.
MG.
A.1
1. U
se g
eom
etric
shap
es, t
heir
mea
sure
s, an
d th
eir p
rope
rtie
s to
desc
ribe
obje
cts
(e.g
., m
odel
ing
a tr
ee tr
unk
or a
hum
an
tors
o as
a c
ylin
der)
. ★
G.M
G.A.
1 U
se g
eom
etric
shap
es, t
heir
mea
sure
s, an
d th
eir p
rope
rtie
s to
desc
ribe
obje
cts.★
For e
xam
ple,
mod
el a
tree
trun
k or
a h
uman
to
rso
as a
cylin
der.
Th
ere
are
no a
sses
smen
t lim
its fo
r thi
s st
anda
rd. T
he e
ntire
stan
dard
is a
sses
sed
in
this
cour
se.
G.M
G.A.
2 2.
App
ly co
ncep
ts o
f den
sity
base
d on
ar
ea a
nd v
olum
e in
mod
elin
g sit
uatio
ns
(e.g
., pe
rson
s per
squa
re m
ile, B
TUs p
er
cubi
c foo
t). ★
G.M
G.A.
3 3.
App
ly g
eom
etric
met
hods
to so
lve
desig
n pr
oble
ms (
e.g.
, des
igni
ng a
n ob
ject
or
stru
ctur
e to
satis
fy p
hysic
al co
nstr
aint
s or
min
imize
cost
; wor
king
with
ty
pogr
aphi
c grid
syst
ems b
ased
on
ratio
s).
★
G.M
G.A.
2 Ap
ply
geom
etric
met
hods
to so
lve
real
-wor
ld p
robl
ems.★
Geom
etric
met
hods
may
inclu
de b
ut a
re n
ot
limite
d to
usin
g ge
omet
ric sh
apes
, the
pr
obab
ility
of a
shad
ed re
gion
, den
sity,
and
de
sign
prob
lem
s.
Ther
e ar
e no
ass
essm
ent l
imits
for t
his
stan
dard
. The
ent
ire st
anda
rd is
ass
esse
d in
th
is co
urse
.
Maj
or C
onte
nt
Su
ppor
ting
Con
tent
83
Notation 2016-17 Standard
2016-17 Scope and Clarification 2017-18 Standard
2017-18 Scope & Clarifications
N-Q.A.1
1. Use units as a w
ay to understand problem
s and to guide the solution of m
ulti-step problems; choose and
interpret units consistently in formulas;
choose and interpret the scale and the origin in graphs and data displays.
There is no additional scope or clarification for this standard.
M1.N.Q
.A.1 Use units as a w
ay to understand problem
s and to guide the solution of m
ulti-step problems;
choose and interpret units consistently in form
ulas; choose and interpret the scale and the origin in graphs and data displays
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
N-Q.A.2
2. Define appropriate quantities for the
purpose of descriptive modeling.
This standard will be assessed in
Math I by ensuring that som
e m
odeling tasks (involving Math I
content or securely held content from
grades 6-8) require the student to create a quantity of interest in the situation being described (i.e., a quantity of interest is not selected for the student by the task). For exam
ple, in a situation involving data, the student m
ight autonomously
decide that a measure of center is
a key variable in a situation, and then choose to w
ork with the
mean.
M1.N.Q
.A.2 Identify, interpret, and justify appropriate quantities for the purpose of descriptive m
odeling.
Clarification: D
escriptive m
odeling refers to understanding and interpreting graphs; identifying extraneous inform
ation; choosing appropriate units; etc. Tasks are lim
ited to linear or exponential equations w
ith integer exponents.
N-Q.A.3
3. Choose a level of accuracy,
appropriate to limitations on
measurem
ent when reporting quantities.
There is no additional scope or clarification for this standard.
M1.N.Q
.A.3 Choose a level of
accuracy appropriate to limitations on
measurem
ent when reporting
quantities.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
A-SSE.A.1 1.Interpret expressions that represent a quantity in term
s of its context. ★
a. Interpret parts of an expression, such as term
s, factors, and coefficients. b. Interpret com
plicated expressions by view
ing 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.
i) Tasks are limited to exponential
expressions, including related num
erical expressions.
M1.A.SSE.A.1 Interpret expressions
that represent a quantity in terms of
its context. ★ a. Interpret parts of an expression, such as term
s, factors, and coefficients. b. Interpret com
plicated expressions by view
ing 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
. Tasks are lim
ited to linear and exponential expressions, including related num
erical expressions
84
A-SS
E.B.
33.
Cho
ose
and
prod
uce
an e
quiv
alen
tfo
rm o
f an
expr
essi
on to
reve
al a
ndex
plai
n pr
oper
ties
of th
e qu
antit
yre
pres
ente
d by
the
expr
essi
on.★
c.
Us e
the
prop
ertie
s of
exp
onen
ts to
trans
form
exp
ress
ions
for e
xpon
entia
lfu
nctio
ns. F
or e
xam
ple
the
expr
ess i
on1.
15t c
an b
e re
writ
ten
as (1
.151/
12)12
t ≈1.
01212
t to re
veal
the
appr
oxim
ate
equi
v ale
nt m
onth
ly in
tere
st ra
te if
the
annu
al ra
te is
15%
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
A.SS
E.B.
2 C
hoos
e an
d pr
oduc
e an
equ
ival
ent f
orm
of a
n ex
pres
sion
to
reve
al a
nd e
xpla
in p
rope
rties
of t
he
quan
tity
repr
esen
ted
by th
e ex
pres
sion
. ★
a.U
se th
e pr
oper
ties
of e
xpon
ents
tore
writ
e ex
pone
ntia
l exp
ress
ions
.
For
M1.
A.S
SE.
B.2
a: F
or
exam
ple,
the
grow
th o
f ba
cter
ia c
an b
e m
odel
ed b
y ei
ther
f(t
) = 3
(t+2)
or
g(t)
= 9
(3t )
bec
ause
the
expr
essi
on 3
(t+2)
can
be
rew
ritte
n as
(3t )
(32 )
= 9
(3t ).
Task
s ha
ve a
real
-wor
ld
cont
ext.
As
desc
ribed
in th
e st
anda
rd, t
here
is a
n in
terp
lay
betw
een
the
mat
hem
atic
al
stru
ctur
e of
the
expr
essi
on
and
the
stru
ctur
e of
the
situ
atio
n su
ch th
at c
hoos
ing
and
prod
ucin
g an
equ
ival
ent
form
of t
he e
xpre
ssio
n re
veal
s so
met
hing
abo
ut th
e si
tuat
ion.
A-CE
D.A.
11.
Cre
ate
equa
tions
and
ineq
ualit
ies
inon
e va
riabl
e an
d us
e th
em to
sol
vepr
oble
ms.
Incl
ude
equa
tions
aris
ing
from
linea
r and
qua
drat
ic fu
nctio
ns, a
ndsi
mpl
e ra
tiona
l and
exp
onen
t ial
f unc
tions
.
i)Ta
sks
are
limite
d to
line
ar o
rex
pone
ntia
l equ
atio
ns w
ith in
tege
rex
pone
nts.
ii) T
asks
hav
e a
real
-w
orld
con
text
. iii)
In th
e lin
ear
case
, tas
ks h
ave
mor
e of
the
hal lm
arks
of m
odel
ing
as a
mat
hem
atic
al p
ract
ice
(less
defin
ed ta
sks,
mor
e of
the
mod
elin
g cy
cle,
etc
.).
M1.
A.CE
D.A.
1 C
reat
e eq
uatio
ns a
nd
ineq
ualit
ies
in o
ne v
aria
ble
and
use
them
to s
olve
pro
blem
s.
i)Ta
sks
are
limite
d to
line
aror
exp
onen
tial e
quat
ions
with
int e
ger e
xpon
ents
.ii)
Task
s ha
ve a
real
-wor
ldc o
ntex
t.iii)
In th
e lin
ear c
ase,
task
sha
ve m
ore
of th
e ha
llmar
ks o
fm
odel
ing
as a
mat
hem
atic
alpr
actic
e (le
ss d
efin
ed ta
sks,
mor
e of
the
mod
elin
g cy
cle,
etc.
).
A-CE
D.A.
22.
Cre
ate
equa
tions
in tw
o or
mor
eva
riabl
es to
repr
esen
t rel
atio
nshi
psbe
twee
n qu
antit
ies;
gra
ph e
quat
ions
on
coor
dina
te a
xes
with
labe
ls a
nd s
cale
s.
i)Ta
sks
are
limite
d to
line
areq
uatio
ns ii
) Tas
ks h
ave
a re
al-
wor
ld c
onte
xt. i
ii) T
asks
hav
e th
eha
llmar
ks o
f mod
elin
g as
am
athe
mat
ical
pra
ctic
e (le
ssde
f ined
task
s, m
ore
of th
em
odel
ing
cycl
e, e
tc.).
M1.
A.CE
D.A.
2 C
reat
e eq
uatio
ns in
tw
o or
mor
e va
riabl
es to
repr
esen
t re
latio
nshi
ps b
etw
een
quan
titie
s;
grap
h eq
uatio
ns w
ith tw
o va
riabl
es o
n co
ordi
nate
axe
s w
ith la
bels
and
sc
ales
.
i)Ta
sks
are
limite
d to
line
areq
uatio
nsii)
Task
s ha
ve a
real
-wor
ldc o
ntex
t.iii)
Task
s ha
ve th
e ha
llmar
ksof
mod
elin
g as
am
athe
mat
ical
pra
ctic
e (le
ssde
fined
task
s, m
ore
of th
em
odel
ing
cycl
e, e
tc.).
85
A-CED.A.3 3. R
epresent constraints by equations or inequalities, and by system
s of equations and/or inequalities, and interpret solutions as viable or nonviable options in a m
odeling context. For exam
ple, represent inequalities describing nutritional and cost constraints on com
binations of different foods.
There is no additional scope or clarification for this standard.
M1.A.CED.A.3 R
epresent constraints by equations or inequalities and by system
s of equations and/or inequalities, and interpret solutions as viable or nonviable options in a m
odeling context.
For example, represent
inequalities describing nutritional and cost constraints on com
binations of different foods. There are no assessm
ent lim
its for this standard. The entire standard is assessed in this course.
A-CED.A.4 4. R
earrange formulas to highlight a
quantity of interest, using the same
reasoning as in solving equations. For exam
ple, rearrange Ohm
’s law V = IR
to highlight resistance R
.
i) Tasks are limited to linear
equations ii) Tasks have a real-w
orld context.
M1.A.CED.A.4 R
earrange formulas
to highlight a quantity of interest, using the sam
e reasoning as in solving equations.
i) Tasks are limited to linear
equations. ii) Tasks have a real-w
orld context.
A-REI.A.3 3. Solve linear equations and inequalities in one variable, including equations w
ith coefficients represented by letters.
There is no additional scope or clarification for this standard.
M1.A.REI.A.1 Solve linear equations
and inequalities in one variable, including equations w
ith coefficients represented by letters.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
A-REI.B.5 5. Prove that, given a system
of two
equations in two variables, replacing one
equation by the sum of that equation and
a multiple of the other produces a
system w
ith the same solutions.
There is no additional scope or clarification for this standard.
A-REI.B.6 6. Solve system
s of linear equations exactly and approxim
ately (e.g., with
graphs), focusing on pairs of linear equations in tw
o variables.
There is no additional scope or clarification for this standard.
M1.A.REI.B.2 W
rite and solve a system
of linear equations in context. S
olve systems both
algebraically and graphically. S
ystems are lim
ited to at m
ost two equations in tw
o variables.
A-REI.C.10 10. U
nderstand that the graph of an equation in tw
o variables is the set of all its solutions plotted in the coordinate plane, often form
ing a curve (which
could be a line).
There is no additional scope or clarification for this standard.
M1.A.REI.C.3 U
nderstand that the graph of an equation in tw
o variables is the set of all its solutions plotted in the coordinate plane, often form
ing a curve (w
hich could be a line).
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
A-REI.C.11 11. Explain w
hy the x-coordinates of the points w
here the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find
i) Tasks that assess conceptual understanding of the indicated concept m
ay involve any of the function types m
entioned in the
M1.A.REI.C.4 Explain w
hy the x-coordinates of the points w
here the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of
Include cases where f(x)
and/or g(x) are linear, absolute value, and exponential functions. For
86
the
solu
tions
app
roxi
mat
ely,
e.g
., us
ing
tech
nolo
gy to
gra
ph th
e fu
nctio
ns, m
ake
tabl
es o
f val
ues,
or f
ind
succ
essi
ve
appr
oxim
atio
ns. I
nclu
de c
ases
whe
re f(
x)
and/
or g
(x) a
re li
near
, pol
ynom
ial,
ratio
nal,
abso
lute
val
ue, e
xpon
entia
l, an
d lo
garit
hmic
func
tions
.★
stan
dard
exc
ept e
xpon
entia
l and
lo
garit
hmic
func
tions
. ii)
Find
ing
the
solu
tions
app
roxi
mat
ely
is
limite
d to
cas
es w
here
f(x)
and
g(
x) a
re p
olyn
omia
l.
the
equa
tion
f(x) =
g(x
); fin
d th
e ap
prox
imat
e so
lutio
ns u
sing
te
chno
logy
. ★
exam
ple:
f(
x) =
3x
+ 5.
i)
Task
s th
at a
sses
s co
ncep
tual
und
erst
andi
ng o
f th
e in
dica
ted
conc
ept m
ay
invo
lve
any
of th
e fu
nctio
n ty
pes
men
tione
d in
the
stan
dard
exc
ept e
xpon
entia
l an
d lo
garit
hmic
func
tions
. ii)
Fin
ding
the
solu
tions
ap
prox
imat
ely
is li
mite
d to
ca
ses
whe
re f(
x) a
nd g
(x) a
re
poly
nom
ial.
iii) T
asks
are
lim
ited
to li
near
an
d ab
solu
te v
alue
func
tions
.
A-RE
I.C.1
2 12
. Gra
ph th
e so
lutio
ns to
a li
near
in
equa
lity
in tw
o va
riabl
es a
s a
halfp
lane
(e
xclu
ding
the
boun
dary
in th
e ca
se o
f a
stric
t ine
qual
ity),
and
grap
h th
e so
lutio
n se
t to
a sy
stem
of l
inea
r ine
qual
ities
in
two
varia
bles
as
the
inte
rsec
tion
of th
e co
rresp
ondi
ng h
alf-
plan
es.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
A.RE
I.C.5
Gra
ph th
e so
lutio
ns to
a
linea
r ine
qual
ity in
two
varia
bles
as
a ha
lf-pl
ane
(exc
ludi
ng th
e bo
unda
ry
in th
e ca
se o
f a s
trict
ineq
ualit
y), a
nd
grap
h th
e so
lutio
n se
t to
a sy
stem
of
linea
r ine
qual
ities
in tw
o va
riabl
es a
s th
e in
ters
ectio
n of
the
corre
spon
ding
ha
lf-pl
anes
.
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
F-IF
.A.1
1.
Und
erst
and
that
a fu
nctio
n fro
m o
ne
set (
calle
d th
e do
mai
n) to
ano
ther
set
(c
alle
d th
e ra
nge)
ass
igns
to e
ach
elem
ent o
f the
dom
ain
exac
tly o
ne
elem
ent o
f the
rang
e. If
f is
a fu
nctio
n an
d x
is a
n el
emen
t of
its
dom
ain,
then
f(x)
den
otes
the
outp
ut o
f f c
orre
spon
ding
to th
e in
put x
. Th
e gr
aph
of f
is th
e gr
aph
of th
e eq
uatio
n
y =
f(x).
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
F.IF
.A.1
Und
erst
and
that
a
func
tion
from
one
set
(cal
led
the
dom
ain)
to a
noth
er s
et (c
alle
d th
e ra
nge)
ass
igns
to e
ach
elem
ent o
f the
do
mai
n ex
actly
one
ele
men
t of t
he
rang
e. If
f is
a fu
nctio
n an
d x
is a
n el
emen
t of i
ts d
omai
n, th
en f(
x)
deno
tes
the
outp
ut o
f f c
orre
spon
ding
to
the
inpu
t x. T
he g
raph
of f
is th
e gr
aph
of th
e eq
uatio
n y
= f(x
).
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
F-IF
.A.2
2.
Use
func
tion
nota
tion,
eva
luat
e fu
nctio
ns fo
r inp
uts
in th
eir d
omai
ns, a
nd
inte
rpre
t sta
tem
ents
that
use
func
tion
no
tatio
n in
term
s of
a c
onte
xt.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
F.IF
.A.2
Use
func
tion
nota
tion,
ev
alua
te fu
nctio
ns fo
r inp
uts
in th
eir
dom
ains
, and
inte
rpre
t sta
tem
ents
th
at u
se fu
nctio
n no
tatio
n in
term
s of
a
cont
ext.
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
87
F-IF.A.33.R
ecognize that sequences arefunctions, som
etimes defined
recursively, whose dom
ain is a subset ofthe integers. For exam
ple, the Fibonaccisequence is defined recursively by f(0) =f(1) = 1,f(n+1) = f(n) +f(n-1) for n ≥ 1.
There is no additional scope or clarification for this standard.
F-IF.B.44.For a function that m
odels arelationship betw
een two quantities,
interpret key features of graphs andt ables in term
s of the quantities, andsketch graphs show
ing key featuresgiven a verbal description of t her elationship. K
ey features include:intercepts; intervals w
here the function isincreasing, decreasing, positive, ornegative; relative m
aximum
s andm
inimum
s; symm
etries; end behavior;and periodicity. ★
i)Tasks have a real-world context.
ii)Tasks are limited to linear
functions, square root functions,cube root functions, piecew
ise-defined functions (including st epfunctions and absolute val uefunctions), and exponentialfunctions w
ith domains in t he
integers. The function types list edhere are the sam
e as those listedin the M
ath I column for standards
F-IF.6 and F-IF.9.
M1.F.IF.B.3 For a function that
models a relationship betw
een two
quantities, interpret key features of graphs and tables in term
s of the quantities, and sketch graphs show
ing key features given a verbal description of the relationship. ★
Key features include:
intercepts; intervals where
the function is increasing, decreasing, positive, or negative; relative m
aximum
s and m
inimum
s; symm
etries; and end behavior.
i)Tasks have a real-wor ld
c ontext.ii)Tasks are lim
ited to linearfunctions, absolute value,and exponential functionsw
ith domains in the integers.
F-IF.B.55.R
elate the domain of a function to its
graph and, where applicable, to the
quantitative relationship it describes. Forexam
ple, if the function h(n) gives t henum
ber of person-hours it takes toassem
ble n engines in a factory, t hent he positive integers w
ould be anappropriate dom
ain for the function. ★
i)Tasks have a real-world context.
ii)Tasks are limited to linear
functions, square root functions,cube root functions, piecew
ise-defined functions (including stepfunctions and absolute val uef unctions), and exponentialfunctions w
ith domains in t he
i ntegers.
M1.F.IF.B.4 R
elate 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
person-hours it takes to assem
ble n engines in a factory, then the positive integers w
ould be an appropriate dom
ain for the function.
i)Tasks have a real-wor ld
c ontext.ii)Tasks are lim
ited to linearfunctions, piecew
ise functions(including step functions andabsolute value functions),and exponential functionsw
ith domains in the integers.
88
F-IF
.B.6
6.
Cal
cula
te a
nd in
terp
ret t
he a
vera
ge
rate
of c
hang
e of
a fu
nctio
n (p
rese
nted
sy
mbo
lical
ly o
r as
a ta
ble)
ove
r a
spec
ified
inte
rval
. Est
imat
e th
e ra
te o
f ch
ange
from
a g
raph
.★
i) Ta
sks
have
a re
al-w
orld
con
text
. ii)
Tas
ks a
re li
mite
d to
line
ar
func
tions
, squ
are
root
func
tions
, cu
be ro
ot fu
nctio
ns, p
iece
wis
e-de
fined
func
tions
(inc
ludi
ng s
tep
func
tions
and
abs
olut
e va
lue
func
tions
), an
d ex
pone
ntia
l fu
nctio
ns w
ith d
omai
ns in
the
inte
gers
. The
func
tion
type
s lis
ted
here
are
the
sam
e as
thos
e lis
ted
in th
e M
ath
I col
umn
for s
tand
ards
F-
IF.4
and
F-IF
.9.
M1.
F.IF
.B.5
Cal
cula
te a
nd in
terp
ret
the
aver
age
rate
of c
hang
e of
a
func
tion
(pre
sent
ed s
ymbo
lical
ly o
r as
a ta
ble)
ove
r a s
peci
fied
inte
rval
. Es
timat
e th
e ra
te o
f cha
nge
from
a
grap
h. ★
i) Ta
sks
have
a re
al-w
orld
co
ntex
t. ii)
Tas
ks a
re li
mite
d to
line
ar
func
tions
, pie
cew
ise
func
tions
(in
clud
ing
step
func
tions
and
ab
solu
te v
alue
func
tions
), an
d ex
pone
ntia
l fun
ctio
ns
with
dom
ains
in th
e in
tege
rs.
F-IF
.C.7
7.
Gra
ph fu
nctio
ns e
xpre
ssed
sy
mbo
lical
ly a
nd s
how
key
feat
ures
of
the
grap
h, b
y ha
nd in
sim
ple
case
s an
d us
ing
tech
nolo
gy fo
r mor
e co
mpl
icat
ed
case
s.★
a. G
raph
line
ar a
nd q
uadr
atic
func
tions
an
d sh
ow in
terc
epts
, max
ima,
and
m
inim
a.
i) Ta
sks
are
limite
d to
line
ar
func
tions
. M
1.F.
IF.C
.6 G
raph
func
tions
ex
pres
sed
sym
bolic
ally
and
sho
w k
ey
feat
ures
of t
he g
raph
, by
hand
and
us
ing
tech
nolo
gy.
a. G
raph
line
ar a
nd q
uadr
atic
fu
nctio
ns a
nd s
how
inte
rcep
ts,
max
ima,
and
min
ima.
Task
s ar
e lim
ited
to li
near
fu
nctio
ns.
F-IF
.C.9
9.
Com
pare
pro
perti
es o
f tw
o fu
nctio
ns
each
repr
esen
ted
in a
diff
eren
t way
(a
lgeb
raic
ally
, gra
phic
ally
, num
eric
ally
in
tabl
es, o
r by
verb
al d
escr
iptio
ns).
For
exam
ple,
giv
en a
gra
ph o
f one
qua
drat
ic
func
tion
and
an a
lgeb
raic
exp
ress
ion
for
anot
her,
say
whi
ch h
as th
e la
rger
m
axim
um.
i) Ta
sks
have
a re
al-w
orld
con
text
. ii)
Tas
ks a
re li
mite
d to
line
ar
func
tions
, squ
are
root
func
tions
, cu
be ro
ot fu
nctio
ns, p
iece
wis
e-de
fined
func
tions
(inc
ludi
ng s
tep
func
tions
and
abs
olut
e va
lue
func
tions
), an
d ex
pone
ntia
l fu
nctio
ns w
ith d
omai
ns in
the
inte
gers
. The
func
tion
type
s lis
ted
here
are
the
sam
e as
thos
e lis
ted
in th
e M
ath
I col
umn
for s
tand
ards
F-
IF.4
and
F-IF
.6
M1.
F.IF
.C.7
Com
pare
pro
perti
es o
f tw
o fu
nctio
ns e
ach
repr
esen
ted
in a
di
ffere
nt w
ay (a
lgeb
raic
ally
, gr
aphi
cally
, num
eric
ally
in ta
bles
, or
by v
erba
l des
crip
tions
). .
i) Ta
sks
have
a re
al-w
orld
co
ntex
t. ii)
Tas
ks a
re li
mite
d to
line
ar
func
tions
, pie
cew
ise
func
tions
(in
clud
ing
step
func
tions
and
ab
solu
te v
alue
func
tions
), an
d ex
pone
ntia
l fun
ctio
ns
with
dom
ains
in th
e in
tege
rs
F-BF
.A.1
1.
W
rite
a fu
nctio
n th
at d
escr
ibes
a
rela
tions
hip
betw
een
two
quan
titie
s.★
a.
D
eter
min
e an
exp
licit
expr
essi
on, a
re
curs
ive
proc
ess,
or s
teps
for
calc
ulat
ion
from
a c
onte
xt.
i) Ta
sks
have
a re
al-w
orld
co
ntex
t. ii)
Ta
sks
are
limite
d to
line
ar
func
tions
and
exp
onen
tial
func
tions
with
dom
ains
in th
e in
tege
rs.
M1.
F.BF
.A.1
Writ
e a
func
tion
that
de
scrib
es a
rela
tions
hip
betw
een
two
quan
titie
s.★
a. D
eter
min
e an
exp
licit
expr
essi
on, a
recu
rsiv
e pr
oces
s, o
r st
eps
for c
alcu
latio
n fro
m a
con
text
.
i) Ta
sks
have
a re
al-w
orld
co
ntex
t. ii)
Tas
ks a
re li
mite
d to
line
ar
func
tions
and
exp
onen
tial
func
tions
with
dom
ains
in th
e in
tege
rs.
89
F-BF.A.2 2. W
rite arithmetic and geom
etric sequences both recursively and w
ith an explicit form
ula, use them to m
odel situations, and translate betw
een the two
forms. ★
There is no additional scope or clarification for this standard.
M1.F.BF.A.2 W
rite arithmetic and
geometric sequences w
ith an explicit form
ula and use them to m
odel situations. ★
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
F-LE.A.1 1. D
istinguish between situations that
can be modeled w
ith linear functions and w
ith exponential functions. a. Prove that linear functions grow
by equal differences over equal intervals, and that exponential functions grow
by equal factors over equal intervals. b. R
ecognize situations in which one
quantity changes at a constant rate per unit interval relative to another. c. R
ecognize situations in which a
quantity grows or decays by a constant
percent rate per unit interval relative to another.
There is no additional scope or clarification for this standard.
M1.F.LE.A.1 D
istinguish between
situations that can be modeled w
ith linear functions and w
ith exponential functions. a. R
ecognize that linear functions grow
by equal differences over equal intervals and that exponential functions grow
by equal factors over equal intervals. b. R
ecognize situations in which one
quantity changes at a constant rate per unit interval relative to another. c. R
ecognize situations in which a
quantity grows or decays by a
constant factor per unit interval relative to another.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
F-LE.A.2 2. C
onstruct linear and exponential functions, including arithm
etic and geom
etric sequences, given a graph, a description of a relationship, or tw
o input-output pairs (include reading these from
a table).
There is no additional scope or clarification for this standard.
M1.F.LE.A.2 C
onstruct linear and exponential functions, including arithm
etic and geometric sequences,
given a graph, a table, a description of a relationship, or input-output pairs.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
F-LE.A.3 3. O
bserve using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (m
ore generally) as a polynom
ial function.
There is no additional scope or clarification for this standard.
M1.F.LE.A.3 O
bserve using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.
Tasks are limited linear and
exponential functions.
F-LE.B.5 5. Interpret the param
eters in a linear or exponential function in term
s of a context.
There is no additional scope or clarification for this standard.
M1.F.LE.B.4 Interpret the param
eters in a linear or exponential function in term
s of a context.
For example, the total cost of
an electrician who charges 35
dollars for a house call and 50 dollars per hour w
ould be expressed as the function y = 50x + 35. If the rate w
ere raised to 65 dollars per hour,
90
desc
ribe
how
the
func
tion
wou
ld c
hang
e.
Task
s ha
ve a
real
-wor
ld
cont
ext.
G-C
O.A
.1
1. K
now
pre
cise
def
initi
ons
of a
ngle
, ci
rcle
, pe
rpen
dicu
lar l
ine,
par
alle
l lin
e,
and
line
segm
ent,
base
d on
the
unde
fined
not
ions
of p
oint
, lin
e, d
ista
nce
alon
g a
line,
and
dis
tanc
e ar
ound
a
circ
ular
arc
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
G.C
O.A
.1 K
now
pre
cise
de
finiti
ons
of a
ngle
, circ
le,
perp
endi
cula
r lin
e, p
aral
lel l
ine,
and
lin
e se
gmen
t, ba
sed
on th
e un
defin
ed
notio
ns o
f poi
nt, l
ine,
pla
ne, d
ista
nce
alon
g a
line,
and
dis
tanc
e ar
ound
a
circ
ular
arc
.
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
G-C
O.A
.2
2. R
epre
sent
tran
sfor
mat
ions
in th
e pl
ane
usin
g, e
.g.,
trans
pare
ncie
s an
d ge
omet
ry s
oftw
are;
des
crib
e tra
nsfo
rmat
ions
as
func
tions
that
take
po
ints
in th
e pl
ane
as in
puts
and
giv
e ot
her p
oint
s as
out
puts
. C
ompa
re tr
ansf
orm
atio
ns th
at p
rese
rve
dist
ance
and
ang
le to
thos
e th
at d
o no
t (e
.g.,
trans
latio
n ve
rsus
hor
izon
tal
stre
tch)
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
G.C
O.A
.2 R
epre
sent
tra
nsfo
rmat
ions
in th
e pl
ane
in
mul
tiple
way
s, in
clud
ing
tech
nolo
gy.
Des
crib
e tra
nsfo
rmat
ions
as
func
tions
th
at ta
ke p
oint
s in
the
plan
e (p
re-
imag
e) a
s in
puts
and
giv
e ot
her
poin
ts (i
mag
e) a
s ou
tput
s. C
ompa
re
trans
form
atio
ns th
at p
rese
rve
dist
ance
and
ang
le m
easu
re to
thos
e th
at d
o no
t (e.
g., t
rans
latio
n ve
rsus
ho
rizon
tal s
tretc
h).
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
G-C
O.A
.3
3. G
iven
a re
ctan
gle,
par
alle
logr
am,
trape
zoid
, or r
egul
ar p
olyg
on, d
escr
ibe
the
rota
tions
and
refle
ctio
ns th
at c
arry
it
onto
itse
lf.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
G.C
O.A
.3 G
iven
a re
ctan
gle,
pa
ralle
logr
am, t
rape
zoid
, or r
egul
ar
poly
gon,
des
crib
e th
e ro
tatio
ns a
nd
refle
ctio
ns th
at c
arry
the
shap
e on
to
itsel
f.
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
G-C
O.A
.4
4. D
evel
op d
efin
ition
s of
rota
tions
, re
flect
ions
, and
tran
slat
ions
in te
rms
of
angl
es, c
ircle
s, p
erpe
ndic
ular
line
s,
para
llel l
ines
, and
line
seg
men
ts.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
G.C
O.A
.4 D
evel
op d
efin
ition
s of
ro
tatio
ns, r
efle
ctio
ns, a
nd tr
ansl
atio
ns
in te
rms
of a
ngle
s, c
ircle
s,
perp
endi
cula
r lin
es, p
aral
lel l
ines
, and
lin
e se
gmen
ts.
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
G-C
O.A
.5
5. G
iven
a g
eom
etric
figu
re a
nd a
ro
tatio
n, re
flect
ion,
or t
rans
latio
n, d
raw
th
e tra
nsfo
rmed
figu
re u
sing
, e.g
., gr
aph
pape
r, tra
cing
pap
er, o
r geo
met
ry
softw
are.
Spe
cify
a s
eque
nce
of
trans
form
atio
ns th
at w
ill ca
rry a
giv
en
figur
e on
to a
noth
er.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
G.C
O.A
.5 G
iven
a g
eom
etric
fig
ure
and
a rig
id m
otio
n, d
raw
the
imag
e of
the
figur
e in
mul
tiple
way
s,
incl
udin
g te
chno
logy
. Spe
cify
a
sequ
ence
of r
igid
mot
ions
that
will
ca
rry a
giv
en fi
gure
ont
o an
othe
r.
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
91
G-C0.B.6
6. Use geom
etric descriptions of rigid m
otions 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.
There is no additional scope or clarification for this standard.
M1.G
.CO.B.6 U
se 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 determ
ine inform
ally if they are congruent.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
G-C0.B.7
7. Use the definition of congruence in
terms of rigid m
otions to show that tw
o triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
There is no additional scope or clarification for this standard.
M1.G
.CO.B.7 U
se the definition of congruence in term
s of rigid motions
to show that tw
o triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
G-CO
.B.8 8. Explain how
the criteria for triangle congruence (ASA, SAS, and SSS) follow
from
the definition of congruence in term
s of rigid motions.
There is no additional scope or clarification for this standard.
M1.G
.CO.B.8 Explain how
the criteria for triangle congruence (ASA, SAS, AAS, and SSS) follow
from the
definition of congruence in terms of
rigid motions.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
G-CO
.C.9 9. Prove theorem
s about lines and angles. Theorem
s include: vertical angles are congruent; w
hen a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segm
ent are exactly those equidistant from
the segment’s
endpoints.
There is no additional scope or clarification for this standard.
M1.G
.CO.C.9 Prove theorem
s about lines and angles.
Proving includes, but is not
limited to, com
pleting partial proofs; constructing tw
o-colum
n or paragraph proofs; using transform
ations to prove theorem
s; analyzing proofs; and critiquing com
pleted proofs. Theorem
s include but are not lim
ited to: vertical angles are congruent; w
hen a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segm
ent are exactly those equidistant from
the segm
ent’s endpoints.
92
G-C
O.C
.10
10.P
rove
theo
rem
s ab
out t
riang
les.
Theo
rem
s in
clud
e: m
easu
res
of in
terio
ran
gles
of a
tria
ngle
sum
to 1
80°;
bas
ean
gles
of i
sosc
eles
tria
ngle
s ar
ec o
ngru
ent;
the
segm
ent j
oini
ngm
idpo
ints
of t
wo
side
s of
a tr
iang
le is
para
llel t
o th
e th
ird s
ide
and
half
t he
leng
t h; t
he m
edia
ns o
f a tr
iang
le m
eet a
ta
poin
t.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
G.C
O.C
.10
Prov
e th
eore
ms
abou
t tri
angl
es.
.
Pro
ving
incl
udes
, but
is n
ot
limite
d to
, com
plet
ing
parti
al
proo
fs; c
onst
ruct
ing
two-
colu
mn
or p
arag
raph
pro
ofs;
us
ing
trans
form
atio
ns to
pr
ove
theo
rem
s; a
naly
zing
pr
oofs
; and
crit
iqui
ng
com
plet
ed p
roof
s.
Theo
rem
s in
clud
e bu
t are
not
lim
ited
to: m
easu
res
of
inte
rior a
ngle
s of
a tr
iang
le
sum
to 1
80°;
bas
e an
gles
of
isos
cele
s tri
angl
es a
re
cong
ruen
t; th
e se
gmen
t jo
inin
g m
idpo
ints
of t
wo
side
s of
a tr
iang
le is
par
alle
l to
the
third
sid
e an
d ha
lf th
e le
ngth
; th
e m
edia
ns o
f a tr
iang
le
mee
t at a
poi
nt.
G-C
O.C
.11
11.P
rove
theo
rem
s ab
out
para
llelo
gram
s. T
heor
ems
incl
ude:
oppo
site
sid
es a
re c
ongr
uent
, opp
osite
angl
es a
re c
ongr
uent
, the
dia
gona
ls o
f apa
ralle
logr
am b
isec
t eac
h ot
her,
and
conv
erse
ly, r
ecta
ngle
s ar
epa
ralle
logr
ams
with
con
grue
nt d
iago
nals
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
G.C
O.C
.11
Prov
e th
eore
ms
abou
t pa
ralle
logr
ams.
P
rovi
ng in
clud
es, b
ut is
not
lim
ited
to, c
ompl
etin
g pa
rtial
pr
oofs
; con
stru
ctin
g tw
o-co
lum
n or
par
agra
ph p
roof
s;
usin
g tra
nsfo
rmat
ions
to
prov
e th
eore
ms;
ana
lyzi
ng
proo
fs; a
nd c
ritiq
uing
co
mpl
eted
pro
ofs.
Theo
rem
s in
clud
e bu
t are
not
lim
ited
to: o
ppos
ite s
ides
are
co
ngru
ent,
oppo
site
ang
les
are
cong
ruen
t, th
e di
agon
als
of a
par
alle
logr
am b
isec
t ea
ch o
ther
, and
con
vers
ely,
re
ctan
gles
are
par
alle
logr
ams
with
con
grue
nt d
iago
nals
.
S-ID
.A.1
1.R
epre
sent
dat
a w
ith p
lots
on
the
real
num
ber l
ine
(dot
plo
ts, h
isto
gram
s, a
ndbo
x pl
ots)
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M1.
S.ID
.A.1
Rep
rese
nt s
ingl
e or
m
ultip
le d
ata
sets
with
dot
plo
ts,
hist
ogra
ms,
ste
m p
lots
(ste
m a
nd
leaf
), an
d bo
x pl
ots.
Ther
e ar
e no
ass
essm
ent
limits
for t
his
stan
dard
. The
en
tire
stan
dard
is a
sses
sed
in
this
cou
rse.
93
S-ID.A.2 2. U
se statistics appropriate to the shape of the data distribution to com
pare center (m
edian, mean) and spread (interquartile
range, standard deviation) of two or
more different data sets.
There is no additional scope or clarification for this standard.
M1.S.ID.A.2 U
se statistics appropriate to the shape of the data distribution to com
pare center (m
edian, mean) and spread
(interquartile range, standard deviation) of tw
o or more different
data sets.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
S-ID.A.3 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extrem
e data points (outliers).
There is no additional scope or clarification for this standard.
M1.S.ID.A.3 Interpret differences in
shape, center, and spread in the context of the data sets, accounting for possible effects of extrem
e data points (outliers).
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
S-ID.B.5 5. Sum
marize categorical data for tw
o categories in tw
o-way frequency tables.
Interpret relative frequencies in the context of the data (including joint, m
arginal, and conditional relative frequencies). R
ecognize possible associations and trends in the data.
There is no additional scope or clarification for this standard.
S-ID.B.6 6. R
epresent data on two quantitative
variables on a scatter plot, and describe how
the variables are related. a. Fit a function to the data; use functions fitted to data to solve problem
s in the context of the data. U
se given functions or choose a function suggested by the context. E
mphasize linear,
quadratic, and exponential models.
c. Fit a linear function for a scatter plot that suggests a linear association.
i) Tasks have real-world context.
ii) Tasks are limited to linear
functions and exponential functions w
ith domains in the
integers.
M1.S.ID.B.4 R
epresent data on two
quantitative variables on a scatter plot, and describe how
the variables are related. a. Fit a function to the data; use functions fitted to data to solve problem
s in the context of the data. U
se given functions or choose a function suggested by the context. b. Fit a linear function for a scatter plot that suggests a linear association.
i) Tasks have real-world
context. ii) Tasks are lim
ited to linear functions and exponential functions w
ith domains in the
integers.
S-ID.C.7 7. Interpret the slope (rate of change) and the intercept (constant term
) of a linear m
odel in the context of the data.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
M1.S.ID.C.5 Interpret the slope (rate
of change) and the intercept (constant term
) of a linear model in
the context of the data.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
S-ID.C.8 8. C
ompute (using technology) and
interpret the correlation coefficient of a linear fit.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
M1.S.ID.C.6 C
ompute (using
technology) and interpret the correlation coefficient of a linear fit.
There are no assessment
limits for this standard. The
entire standard is assessed in this course.
94
S-ID
.C.9
9.D
istin
guis
h be
twee
n co
rrela
tion
and
c aus
atio
n.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
M1.
S.ID
.C.7
Dis
tingu
ish
betw
een
corre
latio
n an
d ca
usat
ion.
Th
ere
are
no a
sses
smen
t lim
its fo
r thi
s st
anda
rd. T
he
entir
e st
anda
rd is
ass
esse
d in
th
is c
ours
e.
★*
= M
odel
ing
Stan
dard
95
Notation 2016-17 Standard
2016-17 Scope and Clarification 2017-18 Standard
2017-18 Scope & Clarifications
N-RN.A.1 1. Explain how
the definition of the m
eaning 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, w
e define 5
1/3 to be the cube root of 5 because we
want (5
1/3) 3 = 5( 1/3) 3 to hold, so (51/3) 3
must equal 5.
There is no additional scope or clarification for this standard.
M2.N.RN.A.1 Explain how
the definition of the m
eaning of rational exponents follow
s from extending
the properties of integer exponents to those values, allow
ing for a notation for radicals in term
s of rational exponents.
For example, w
e define 5 1/3 to be the cube root of 5 because w
e want (5
1/3) 3 = 5( 1/3) 3 to hold, so (5
1/3) 3 must equal 5.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
N-RN.A.2 2. R
ewrite expressions involving radicals
and rational exponents using the properties of exponents.
There is no additional scope or clarification for this standard.
M2.N.RN.A.2 R
ewrite expressions
involving radicals and rational exponents using the properties of exponents.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
N-RN.A.3 3. Explain w
hy the sum or product of tw
o rational num
bers 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.
There is no additional scope or clarification for this standard.
N-Q.A.2
2. Define appropriate quantities for the
purpose of descriptive modeling.
This standard will be assessed in
Math II by ensuring that som
e m
odeling tasks (involving Math I
content or securely held content from
grades 6-8) require the student to create a quantity of interest in the situation being described (i.e., a quantity of interest is not selected for the student by the task). For exam
ple, in a situation involving data, the student m
ight autonom
ously decide that a measure
of center is a key variable in a situation, and then choose to w
ork w
ith the mean.
M2.N.Q
.A.1 Identify, interpret, and justify appropriate quantities for the purpose of descriptive m
odeling.
Descriptive m
odeling refers to understanding and interpreting graphs; identifying extraneous inform
ation; choosing appropriate units; etc. Tasks are lim
ited to linear or exponential equations w
ith integer exponents.
N-CN.A.1 1. Know
there is a complex num
ber i such that i 2 = –1, and every com
plex num
ber has the form a + bi w
ith a and b real.
There is no additional scope or clarification for this standard.
M2.N.CN.A.1 Know
there is a com
plex number i such that i 2 = –1,
and every complex num
ber has the form
a + bi with a and b real.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
97
N-C
N.A
.22.
Use
the
rela
tion
i 2 =
–1
and
the
com
mut
ativ
e, a
ssoc
iativ
e, a
nd
dist
ribut
ive
prop
ertie
s to
add
, sub
tract
, an
d m
ultip
ly c
ompl
ex n
umbe
rs.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
N.C
N.A
.2 K
now
and
use
the
rela
tion
i 2 =
–1
and
the
com
mut
ativ
e,
asso
ciat
ive,
and
dis
tribu
tive
prop
ertie
s to
add
, sub
tract
, and
m
ultip
ly c
ompl
ex n
umbe
rs.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
co
urse
.
N-C
N.B
.77.
Sol
ve q
uadr
atic
equ
atio
ns w
ith re
al
coef
ficie
nts
that
hav
e co
mpl
ex s
olut
ions
.Th
ere
is n
o ad
ditio
nal s
cope
or
clar
ifica
tion
for t
his
stan
dard
.M
2.N
.CN
.B.3
Sol
ve q
uadr
atic
eq
uatio
ns w
ith re
al c
oeffi
cien
ts th
at
have
com
plex
sol
utio
ns.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
co
urse
.
A-SS
E.A.
11.
Inte
rpre
t exp
ress
ions
that
repr
esen
t a
quan
tity
in te
rms
of it
s co
ntex
t. ★
b. In
terp
ret c
ompl
icat
ed e
xpre
ssio
ns b
y vi
ewin
g on
e or
mor
e of
thei
r par
ts a
s a
sing
le e
ntity
. For
exa
mpl
e, in
terp
ret
P(1
+r) n
as
the
prod
uct o
f P a
nd a
fact
or
not d
epen
ding
on
P.
i) Ta
sks
are
limite
d to
qua
drat
ic
expr
essi
ons.
M2.
A.SS
E.A.
1 In
terp
ret e
xpre
ssio
ns
that
repr
esen
t a q
uant
ity in
term
s of
its
con
text
. ★
a. In
terp
ret c
ompl
icat
ed e
xpre
ssio
ns
by v
iew
ing
one
or m
ore
of th
eir p
arts
as
a s
ingl
e en
tity.
For e
xam
ple,
inte
rpre
t P(1
+ r)
n as
the
prod
uct o
f P a
nd a
fact
or
not d
epen
ding
on
P.
Task
s ar
e lim
ited
to q
uadr
atic
ex
pres
sion
s.
A-SS
E.A.
2 U
se th
e st
ruct
ure
of a
n ex
pres
sion
to
iden
tify
way
s to
rew
rite
it. F
or e
xam
ple,
s
ee
x 4
– y
4 as
(x 2 )
2 –
(y 2 )
2 , th
us re
cogn
izin
g it
as
a di
ffere
nce
of s
quar
es th
at c
an b
e fa
ctor
ed a
s (x
2 –
y 2 )(
x 2
+ y
2 ).
i) Ta
sks
are
limite
d to
qua
drat
ic
and
expo
nent
ial e
xpre
ssio
ns,
incl
udin
g re
late
d nu
mer
ical
ex
pres
sion
s.ii)
Exa
mpl
es: S
ee a
n op
portu
nity
to
rew
rite
a 2
+ 9a
+ 1
4 as
(a+7
)(a+
2).
Rec
ogni
ze 5
3 2
- 47
2 as
a d
iffer
ence
of
squ
ares
and
see
an
oppo
rtuni
ty
to re
writ
e it
in th
e ea
sier
-to-
eval
uate
form
(53+
47)(
53-4
7).
M2.
A.SS
E.A.
2 U
se th
e st
ruct
ure
of
an e
xpre
ssio
n to
iden
tify
way
s to
re
writ
e it.
For e
xam
ple,
reco
gniz
e 5
3 2
- 47
2 as
a d
iffer
ence
of
squa
res
and
see
an
oppo
rtuni
ty to
rew
rite
it in
the
easi
er-to
-eva
luat
e fo
rm
(53
+ 47
) (53
– 4
7).
See
an
oppo
rtuni
ty to
rew
rite
a 2
+ 9a
+ 1
4 as
(a +
7) (
a +
2).
Task
s ar
e lim
ited
to n
umer
ical
ex
pres
sion
s an
d po
lyno
mia
l ex
pres
sion
s in
one
var
iabl
e
A-SS
E.B
.33.
Cho
ose
and
prod
uce
an e
quiv
alen
t fo
rm o
f an
expr
essi
on to
reve
al a
nd
expl
ain
prop
ertie
s of
the
quan
tity
repr
esen
ted
by th
e ex
pres
sion
. ★a.
Fac
tor a
qua
drat
ic e
xpre
ssio
n to
re
veal
the
zero
s of
the
func
tion
it de
fines
.b.
Com
plet
e th
e sq
uare
in a
qua
drat
ic
expr
essi
on to
reve
al th
e m
axim
um o
r m
inim
um v
alue
of t
he fu
nctio
n it
defin
es.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
A.SS
E.B
.3 C
hoos
e an
d pr
oduc
e an
equ
ival
ent f
orm
of a
n ex
pres
sion
to
reve
al a
nd e
xpla
in p
rope
rties
of
the
quan
tity
repr
esen
ted
by th
e ex
pres
sion
. ★a.
Fac
tor a
qua
drat
ic e
xpre
ssio
n to
re
veal
the
zero
s of
the
func
tion
it de
fines
.b.
Com
plet
e th
e sq
uare
in a
qu
adra
tic e
xpre
ssio
n in
the
form
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
Ax
2 + Bx + C
where A = 1 to reveal
the maxim
um or m
inimum
value of the function it defines.
A-APR.A.11.U
nderstand that polynomials form
asystem
analogous to the integers,nam
ely, they are closed under theoperations of addition, subtraction,and m
ultiplication; add, subtract, andm
ultiply polynomials
There is no additional scope or clarification for this standard.
M2.A.APR.A.1 U
nderstand that polynom
ials form a system
analogous to the integers, nam
ely, they are closed under the operations of addition, subtraction, and m
ultiplication; add, subtract, and m
ultiply polynomials.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A-CED.A.11.C
reate equations and inequalities inone variable and use them
to solveproblem
s. Include equations arising fromlinear and quadratic functions, andsim
ple rational and exponentialfunctions.
i)Tasks are limited to quadratic
and exponential equations.ii)Tasks have a real-w
orld context.iii) In sim
pler cases (such asexponential equations w
ith integerexponents), tasks have m
ore of t hehal lm
arks of modeling as a
mathem
atical practice (less definedtasks, m
ore of the modeling cycle,
etc.).
M2.A.CED.A.1 C
reate equations and inequalities in one variable and use them
to solve problems.
Include equations arising from
linear and quadratic functions and rational and exponential functions. Tasks have a real-w
orld context.
A-CED.A.22.C
reate equations in two or m
orevariables to represent relationshipsbetw
een quantities; graph equations onc oordinate axes w
ith labels and scales.
i)Tasks are limited to quadratic
equations.ii)Tasks have a real-w
orld context.iii) Tasks have the hallm
arks ofm
odeling as a mathem
aticalpractice (less defined tasks, m
or eof the m
odeling cycle, etc.).
M2.A.CED.A.2 C
reate equations in tw
o or more variables to represent
relationships between quantities;
graph equations with tw
o variables on coordinate axes w
ith labels and scales.
i)Tasks are limited to quadratic
equationsii)Tasks have a real-w
orldcontext.iii)Tasks have the hallm
arks ofm
odeling as a mathem
atic alpr actice (less defined tasks,m
ore of the modeling cycle,
etc.).
A-CED.A.44.R
earrange formulas to highlight a
quantity of interest, using the same
reasoning as in solving equations. Forexam
ple, rearrange Ohm
’s law V = IR
tohighlight resistance R
.
i)Tasks are limited to quadratic
equations.ii)Tasks have a real-w
orld context.
M2.A.CED.A.3 R
earrange formulas
to highlight a quantity of interest, using the sam
e reasoning as in solving equations.
i)Tasks are limited to quadratic,
square root, cube root, andpiec ew
ise functions.ii) Tasks have a real-w
orld context.
i)Tasks are limited to
quadratic, square root, cuberoot, and piecew
ise functions.ii)Tasks have a real-w
orldcontext.iii)Tasks have the hallm
arks ofm
odeling as a mathem
atic alpr actice (less defined tasks,m
ore of the modeling cycle,
etc.).
A-RE
I.A.1
1.
Exp
lain
eac
h st
ep in
sol
ving
a s
impl
e eq
uatio
n as
follo
win
g fro
m th
e eq
ualit
y of
nu
mbe
rs a
sser
ted
at th
e pr
evio
us s
tep,
st
artin
g fro
m th
e as
sum
ptio
n th
at th
e or
igin
al e
quat
ion
has
a so
lutio
n.
Con
stru
ct a
via
ble
argu
men
t to
just
ify a
so
lutio
n m
etho
d.
i) Ta
sks
are
limite
d to
qua
drat
ic
equa
tions
. M
2.A.
REI.A
.1 E
xpla
in e
ach
step
in
solv
ing
an e
quat
ion
as fo
llow
ing
from
the
equa
lity
of n
umbe
rs
asse
rted
at th
e pr
evio
us s
tep,
st
artin
g fro
m th
e as
sum
ptio
n th
at th
e or
igin
al e
quat
ion
has
a so
lutio
n.
Con
stru
ct a
via
ble
argu
men
t to
just
ify a
sol
utio
n m
etho
d.
Task
s ar
e lim
ited
to li
near
, qu
adra
tic, e
xpon
entia
l eq
uatio
ns w
ith in
tege
r ex
pone
nts,
squ
are
root
, cub
e ro
ot, p
iece
wis
e, a
nd
expo
nent
ial f
unct
ions
.
A-RE
I.B.4
4.
Sol
ve q
uadr
atic
equ
atio
ns in
one
va
riabl
e.
a. U
se th
e m
etho
d of
com
plet
ing
the
squa
re to
tran
sfor
m a
ny q
uadr
atic
eq
uatio
n in
x in
to a
n eq
uatio
n of
the
form
(x
– p
)2 = q
that
has
the
sam
e so
lutio
ns.
Der
ive
the
quad
ratic
form
ula
from
this
fo
rm.
b. S
olve
qua
drat
ic e
quat
ions
by
insp
ectio
n (e
.g.,
for x
2 = 4
9), t
akin
g sq
uare
root
s, c
ompl
etin
g th
e sq
uare
, the
qu
adra
tic fo
rmul
a an
d fa
ctor
ing,
as
appr
opria
te to
the
initi
al fo
rm o
f the
eq
uatio
n. R
ecog
nize
whe
n th
e qu
adra
tic
form
ula
give
s co
mpl
ex s
olut
ions
and
w
rite
them
as
a ±
bi fo
r rea
l num
bers
a
and
b.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
A.RE
I.B.2
Sol
ve q
uadr
atic
eq
uatio
ns a
nd in
equa
litie
s in
one
va
riabl
e.
a. U
se th
e m
etho
d of
com
plet
ing
the
squa
re to
rew
rite
any
quad
ratic
eq
uatio
n in
x in
to a
n eq
uatio
n of
the
form
(x –
p)2 =
q th
at h
as th
e sa
me
solu
tions
. Der
ive
the
quad
ratic
fo
rmul
a fro
m th
is fo
rm.
b. S
olve
qua
drat
ic e
quat
ions
by
insp
ectio
n (e
.g.,
for x
2 = 4
9), t
akin
g sq
uare
root
s, c
ompl
etin
g th
e sq
uare
, kn
owin
g an
d ap
plyi
ng th
e qu
adra
tic
form
ula,
and
fact
orin
g, a
s ap
prop
riate
to th
e in
itial
form
of t
he
equa
tion.
Rec
ogni
ze w
hen
the
quad
ratic
form
ula
give
s co
mpl
ex
solu
tions
and
writ
e th
em a
s a
± bi
for
real
num
bers
a a
nd b
.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
A-RE
I.B.7
7.
Sol
ve a
sim
ple
syst
em c
onsi
stin
g of
a
linea
r equ
atio
n an
d a
quad
ratic
equ
atio
n in
two
varia
bles
alg
ebra
ical
ly a
nd
grap
hica
lly. F
or e
xam
ple,
find
the
poin
ts
of in
ters
ectio
n be
twee
n th
e lin
e y
= –3
x an
d th
e ci
rcle
x2 +
y2 =
3.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
A.RE
I.C.3
Writ
e an
d so
lve
a sy
stem
of l
inea
r equ
atio
ns in
co
ntex
t.
Whe
n so
lvin
g al
gebr
aica
lly,
task
s ar
e lim
ited
to s
yste
ms
of
at m
ost t
hree
equ
atio
ns a
nd
thre
e va
riabl
es. W
ith g
raph
ic
solu
tions
sys
tem
s ar
e lim
ited
to
only
two
varia
bles
.
M2.
A.RE
I.C.4
Sol
ve a
sys
tem
co
nsis
ting
of a
line
ar e
quat
ion
and
a qu
adra
tic e
quat
ion
in tw
o va
riabl
es
alge
brai
cally
and
gra
phic
ally
.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
99
F-IF.B.4 4. For a function that m
odels a relationship betw
een two quantities,
interpret key features of graphs and tables in term
s of the quantities, and sketch graphs show
ing key features given a verbal description of the relationship. K
ey features include: intercepts; intervals w
here the function is increasing, decreasing, positive, or negative; relative m
aximum
s and m
inimum
s; symm
etries; end behavior; and periodicity. ★
I)Tasks have a real-world context.
ii) Tasks are limited to quadratic
and exponential functions. The function types listed here are the sam
e as those listed in Math II
column for standards F-IF.6 and F-
IF.9.
M2.F.IF.A.1 For a function that
models a relationship betw
een two
quantities, interpret key features of graphs and tables in term
s of the quantities and sketch graphs show
ing key features given a verbal description of the relationship. ★
Key features include:
intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative m
aximum
s and m
inimum
s; symm
etries; and end behavior. i) Tasks have a real-w
orld context. ii) Tasks are lim
ited to quadratic, exponential functions w
ith integer exponents, square root, and cube root functions.
F-IF.B.5 5.R
elate the domain of a function to its
graph and, where applicable, to the
quantitative relationship it describes. For exam
ple, if the function h(n) gives the num
ber of person-hours it takes to assem
ble n engines in a factory, then the positive integers w
ould be an appropriate dom
ain for the function. ★
i) Tasks have a real-world context.
ii) Tasks are limited to quadratic
functions.
M2.F.IF.A.2 R
elate 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
person-hours it takes to assem
ble n engines in a factory, then the positive integers w
ould be an appropriate dom
ain for the function. Tasks are lim
ited to quadratic, square root, cube root, piecew
ise, and exponential functions.
F-IF.B.6 6. C
alculate and interpret the average rate of change of a function (presented sym
bolically or as a table) over a specified interval. Estim
ate the rate of change from
a graph. ★
i) Tasks have a real-world context.
ii) Tasks are limited to quadratic
and exponential functions. The function types listed here are the sam
e as those listed in the M
ath II column for standards F-IF.4
and F-IF.9.
M2.F.IF.A.3 C
alculate and interpret the average rate of change of a function (presented sym
bolically or as a table) over a specified interval. Estim
ate the rate of change from a
graph. ★
i) Tasks have a real-world
context. ii) Tasks m
ay involve quadratic, square root, cube root, piecew
ise, and exponential functions.
F-IF.C.7 7. G
raph functions expressed sym
bolically and show key features of
the graph, by hand in simple cases and
using technology for more com
plicated
For F-IF.7a: i) Tasks are lim
ited to quadratic functions.
M2.F.IF.B.4 G
raph functions expressed sym
bolically and show
key features of the graph, by hand and using technology. ★
M2.F.IF.B.4a – Tasks are
limited to quadratic functions.
M2.F.IF.B.4c – Tasks are
case
s.★
a.
Gra
ph li
near
and
qua
drat
ic fu
nctio
nsan
d sh
ow in
terc
epts
, max
ima,
and
min
ima.
b.G
raph
squ
are
root
, cub
e ro
ot, a
ndpi
ecew
ise-
defin
ed fu
nctio
ns, i
nclu
ding
step
func
tions
and
abs
olut
e va
lue
func
tions
.e.
Gra
ph e
xpon
entia
l and
loga
rithm
i cfu
nctio
ns, s
how
ing
inte
rcep
ts a
nd e
ndbe
hav i
or, a
nd tr
igon
omet
ric fu
nctio
ns,
show
ing
perio
d, m
idlin
e, a
nd a
mpl
itude
.
For F
-IF.7
e:
i)Ta
sks
are
limite
d to
exp
onen
tial
func
tions
.
a.G
raph
line
ar a
nd q
uadr
atic
func
tions
and
sho
win
terc
epts
, max
ima,
and
min
ima.
b.G
raph
squ
are
root
, cub
er o
ot, a
nd p
iece
wis
e-de
fined
func
tions
, inc
ludi
ng s
t ep
f unc
tions
and
abs
olut
e va
lue
f unc
tions
.c.
c. G
raph
exp
onen
ti al a
ndl o
garit
hmic
func
tions
,sh
owin
g in
terc
epts
and
end
beha
vior
.
limite
d to
exp
onen
tial f
unct
ions
.
8.W
rite
a fu
nctio
n de
fined
by
anex
pres
sion
in d
iffer
ent b
ut e
quiv
alen
tfo
rms
to re
veal
and
exp
lain
diff
eren
tpr
oper
ties
of th
e fu
nctio
n.a.
Use
the
proc
ess
of fa
ctor
ing
and
c om
plet
ing
the
squa
re in
a q
uadr
atic
func
tion
to s
how
zer
os, e
xtre
me
valu
es,
and
sym
met
ry o
f the
gra
ph, a
nd in
terp
ret
thes
e in
term
s of
a c
onte
xt.
b.U
se th
e pr
oper
ties
of e
xpon
ents
toin
terp
ret e
xpre
ssio
ns fo
r exp
onen
tial
f unc
tions
. For
exa
mpl
e, id
entif
y pe
rcen
tra
te o
f cha
nge
in fu
nctio
ns s
uch
as y
=(1
.02)
t , y
= (0
.97)
t , y
= (1
.01)
12t ,
y =
(1.2
)t/10 ,
and
clas
sify
them
as
repr
esen
ting
expo
nent
ial g
row
th o
rde
cay.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
F.IF
.B.5
Writ
e a
func
tion
defin
ed
by a
n ex
pres
sion
in d
iffer
ent b
ut
equi
vale
nt fo
rms
to re
veal
and
ex
plai
n di
ffere
nt p
rope
rties
of t
he
func
tion.
a.
Use
the
proc
ess
of fa
ctor
ing
and
c om
plet
ing
the
squa
rein
a q
uadr
atic
func
tion
t osh
ow z
eros
, ext
rem
e va
lues
,an
d sy
mm
etry
of t
he g
raph
,an
d in
terp
ret t
hese
in te
rms
of a
con
text
.b.
b. K
now
and
use
the
prop
ertie
s of
exp
onen
ts to
i nte
rpre
t exp
ress
ions
for
expo
nent
ial f
unct
ions
.
For e
xam
ple,
iden
tify
perc
ent
rate
of c
hang
e in
func
tions
su
ch a
s y
= 2x
, y
= (1
/2)x ,
y =
2-x,
y =
(1/2
)-x.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
F-IF
.C.9
9.C
ompa
re p
rope
rties
of t
wo
func
tions
each
repr
esen
ted
in a
diff
eren
t way
(alg
ebra
ical
ly, g
raph
ical
ly, n
umer
ical
ly in
tabl
es, o
r by
verb
al d
escr
iptio
ns).
For
exam
ple,
giv
en a
gra
ph o
f one
qua
drat
icfu
nctio
n an
d an
alg
ebra
ic e
xpre
ssio
n fo
ran
othe
r, sa
y w
hich
has
the
larg
erm
axim
um.
i)Ta
sks
are
limite
d to
on
quad
ratic
and
expo
nent
ial f
unct
ions
. ii)
Task
sdo
not
hav
e a
real
-wor
ld c
onte
xt.
The
func
tion
type
s lis
ted
here
ar e
the
sam
e as
thos
e lis
ted
in t h
eM
ath
II co
lum
n fo
r sta
ndar
ds F
-IF.4
and
F-IF
.6.
M2.
F.IF
.B.6
Com
pare
pro
perti
es o
f tw
o fu
nctio
ns e
ach
repr
esen
ted
in a
di
ffere
nt w
ay (a
lgeb
raic
ally
, gr
aphi
cally
, num
eric
ally
in ta
bles
, or
by v
erba
l des
crip
tions
).
i)Ta
sks
do n
ot h
ave
a re
al-
wor
ld c
onte
xt.
ii)Ta
sks
may
invo
lve
quad
ratic
,sq
uare
root
, cub
e ro
ot,
piec
ewis
e, a
nd e
xpon
entia
lfu
nctio
ns.
101
F-BF.A.1 1. W
rite a function that describes a relationship betw
een two quantities. ★
b. Com
bine standard function types using arithm
etic operations. For exam
ple, 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 m
odel.
For F-BF.1a:
i) Tasks have real-world content.
ii) Tasks may involve linear
functions, quadratic functions, and exponential functions.
M2.F.BF.A.1 W
rite a function that describes a relationship betw
een tw
o quantities. ★ a. D
etermine an explicit expression,
a recursive process, or steps for calculation from
a context. b. C
ombine standard function types
using arithmetic operations.
For M2.F.B
F.A.1a:
i) Tasks have a real-world
context. ii) Tasks m
ay involve linear and quadratic functions.
F-BF.B.3 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. Experim
ent with cases
and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from
their graphs an d algebraic expressions for them
.
i) Identifying 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) is lim
ited to linear and quadratic functions. ii) Experim
enting with
cases and illustrating an explanation of the effects on the graph using technology is lim
ited to linear functions, quadratic functions, square root functions, cube root functions, piecew
ise-defined functions (including step functions and absolute value functions), and exponential functions. iii) Tasks do not involve recognizing even and odd functions.
M2.F.BF.B.2 Identify the effect on
the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experim
ent with cases
and illustrate an explanation of the effects on the graph using technology.
i) Identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), and f(x+k) for specific values of k (both positive and negative) is lim
ited to linear, quadratic, and absolute value functions. ii) E
xperimenting w
ith cases and illustrating an explanation of the effects on the graph using technology is lim
ited to linear, quadratic, square root, cube root, and exponential functions. iii) Tasks do not involve recognizing even and odd functions.
G-SRT.A.1
1. 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 segm
ent is longer or shorter in the ratio given by the scale factor.
There is no additional scope or clarification for this standard.
M2.G
.SRT.A.1 Verify informally the
properties of dilations given by a center and a scale factor.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G-S
RT.A
.2
2. G
iven
two
figur
es, u
se th
e de
finiti
on o
f si
mila
rity
in te
rms
of s
imila
rity
trans
form
atio
ns to
dec
ide
if th
ey a
re
sim
ilar;
expl
ain
usin
g si
mila
rity
trans
form
atio
ns th
e m
eani
ng o
f sim
ilarit
y fo
r tria
ngle
s as
the
equa
lity
of a
ll co
rresp
ondi
ng p
airs
of a
ngle
s an
d th
e pr
opor
tiona
lity
of a
ll co
rresp
ondi
ng p
airs
of
sid
es.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
G.S
RT.A
.2 G
iven
two
figur
es,
use
the
defin
ition
of s
imila
rity
in
term
s of
sim
ilarit
y tra
nsfo
rmat
ions
to
deci
de if
they
are
sim
ilar;
expl
ain
usin
g si
mila
rity
trans
form
atio
ns th
e m
eani
ng o
f sim
ilarit
y fo
r tria
ngle
s as
th
e eq
ualit
y of
all
corre
spon
ding
pa
irs o
f ang
les
and
the
prop
ortio
nalit
y of
all
corre
spon
ding
pa
irs o
f sid
es.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
G-S
RT.A
.3
3. U
se th
e pr
oper
ties
of s
imila
rity
trans
form
atio
ns to
est
ablis
h th
e AA
cr
iterio
n fo
r tw
o tri
angl
es to
be
sim
ilar.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
G.S
RT.A
.3 U
se th
e pr
oper
ties
of
sim
ilarit
y tra
nsfo
rmat
ions
to
esta
blis
h th
e AA
crit
erio
n fo
r tw
o tri
angl
es to
be
sim
ilar.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
G-S
RT.B
.4
4. P
rove
theo
rem
s ab
out t
riang
les.
Th
eore
ms
incl
ude:
a li
ne p
aral
lel t
o on
e si
de o
f a tr
iang
le d
ivid
es th
e ot
her t
wo
prop
ortio
nally
, and
con
vers
ely;
the
Pyt
hago
rean
The
orem
pro
ved
usin
g tri
angl
e si
mila
rity.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
G.S
RT.B
.4 P
rove
theo
rem
s ab
out s
imila
r tria
ngle
s.
Pro
ving
incl
udes
, but
is n
ot
limite
d to
, com
plet
ing
parti
al
proo
fs; c
onst
ruct
ing
two-
colu
mn
or p
arag
raph
pro
ofs;
us
ing
trans
form
atio
ns to
pro
ve
theo
rem
s; a
naly
zing
pro
ofs;
an
d cr
itiqu
ing
com
plet
ed
proo
fs.
Th
eore
ms
incl
ude
but a
re n
ot
limite
d to
: a li
ne p
aral
lel t
o on
e si
de o
f a tr
iang
le d
ivid
es th
e ot
her t
wo
prop
ortio
nally
, and
co
nver
sely
; the
Pyt
hago
rean
Th
eore
m p
rove
d us
ing
trian
gle
sim
ilarit
y.
G-S
RT.B
.5
5. U
se c
ongr
uenc
e an
d si
mila
rity
crite
ria
for t
riang
les
to s
olve
pro
blem
s an
d to
pr
ove
rela
tions
hips
in g
eom
etric
figu
res.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
G.S
RT.B
.5 U
se c
ongr
uenc
e an
d si
mila
rity
crite
ria fo
r tria
ngle
s to
so
lve
prob
lem
s an
d to
just
ify
rela
tions
hips
in g
eom
etric
figu
res.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
G-S
RT.C
.6
6. U
nder
stan
d th
at b
y si
mila
rity,
sid
e ra
tios
in ri
ght t
riang
les
are
prop
ertie
s of
th
e an
gles
in th
e tri
angl
e, le
adin
g to
de
finiti
ons
of tr
igon
omet
ric ra
tios
for
acut
e an
gles
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
G.S
RT.C
.6 U
nder
stan
d th
at b
y si
mila
rity,
sid
e ra
tios
in ri
ght t
riang
les
are
prop
ertie
s of
the
angl
es in
the
trian
gle,
lead
ing
to d
efin
ition
s of
tri
gono
met
ric ra
tios
for a
cute
ang
les.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
103
G.SRT.C.7
7. Explain and use the relationship betw
een the sine and cosine of com
plementary angles.
There is no additional scope or clarification for this standard.
M2.G
.SRT.C.7 Explain and use the relationship betw
een the sine and cosine of com
plementary angles.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G-SRT.C.8
8. Use trigonom
etric ratios and the Pythagorean Theorem
to solve right triangles in applied problem
s. ★
There is no additional scope or clarification for this standard.
M2.G
.SRT.C.8 Solve triangles. ★ a. Know
and use trigonometric ratios
and the Pythagorean Theorem to
solve right triangles in applied problem
s. b. Know
and use the Law of Sines
and the Law of C
osines to solve triangles in applied problem
s. R
ecognize when it is appropriate to
use each.
Am
biguous cases will not be
included in assessment.
G-G
MD.A.1
1. Give an inform
al argument for the
formulas for the circum
ference of a circle, area of a circle, volum
e of a cylinder, pyram
id, and cone. Use
dissection arguments, C
avalieri’s principle, and inform
al limit argum
ents.
There is no additional scope or clarification for this standard.
M2.G
.GM
D.A.1 Give an inform
al argum
ent for the formulas for the
circumference of a circle and the
volume and surface area of a
cylinder, cone, prism, and pyram
id.
Informal argum
ents may
include but are not limited to
using the dissection argument,
applying Cavalieri’s principle,
and constructing informal lim
it argum
ents.
G-G
MD.A.3
3. Use volum
e formulas for cylinders,
pyramids, cones, and spheres to solve
problems. ★
There is no additional scope or clarification for this standard.
M2.G
.GM
D.A.2 Know and use
volume and surface area form
ulas for cylinders, cones, prism
s, pyram
ids, and spheres to solve problem
s. ★
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
S-ID.B.6 6. R
epresent data on two quantitative
variables on a scatter plot, and describe how
the variables are related. a. Fit a function to the data; use functions fitted to data to solve problem
s in the context of the data. U
se given functions or choose a function suggested by the context. E
mphasize linear,
quadratic, and exponential models.
b. Informally assess the fit of a function
by plotting and analyzing residuals.
For S-ID.A.6a:
i) Tasks have real-world context.
ii) Tasks are limited to quadratic
functions. For S-ID
.A.6b: i) Tasks have a real-w
orld context. ii) Tasks are lim
ited to linear functions, quadratic functions, and exponential functions w
ith domains
in the integers.
M2.S.ID.A.1 R
epresent data on two
quantitative variables on a scatter plot, and describe how
the variables are related. a. Fit a function to the data; use functions fitted to data to solve problem
s in the context of the data.
Use given functions or choose
a function suggested by the context. E
mphasize linear,
quadratic, and exponential m
odels. Exponential functions
are limited to those w
ith dom
ains in the integers. Tasks have a real-w
orld context.
S-CP
.A.1
1.
Des
crib
e ev
ents
as
subs
ets
of a
sa
mpl
e sp
ace
(the
set o
f out
com
es)
usin
g ch
arac
teris
tics
(or c
ateg
orie
s) o
f th
e ou
tcom
es, o
r as
unio
ns,
inte
rsec
tions
, or c
ompl
emen
ts o
f oth
er
even
ts (“
or,”
“and
,” “n
ot”).
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
S.CP
.A.1
Des
crib
e ev
ents
as
subs
ets
of a
sam
ple
spac
e (th
e se
t of
out
com
es) u
sing
cha
ract
eris
tics
(or c
ateg
orie
s) o
f the
out
com
es, o
r as
uni
ons,
inte
rsec
tions
, or
com
plem
ents
of o
ther
eve
nts
(“or,”
“a
nd,”
“not
”).
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
S-CP
.A.2
2.
Und
erst
and
that
two
even
ts A
and
B
are
inde
pend
ent i
f the
pro
babi
lity
of A
an
d B
occ
urrin
g to
geth
er is
the
prod
uct
of th
eir p
roba
bilit
ies,
and
use
this
ch
arac
teriz
atio
n to
det
erm
ine
if th
ey a
re
inde
pend
ent.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
S.CP
.A.2
Und
erst
and
that
two
even
ts A
and
B a
re in
depe
nden
t if
the
prob
abili
ty o
f A a
nd B
occ
urrin
g to
geth
er is
the
prod
uct o
f the
ir pr
obab
ilitie
s, a
nd u
se th
is
char
acte
rizat
ion
to d
eter
min
e if
they
ar
e in
depe
nden
t.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
S-CP
.A.3
3.
Und
erst
and
the
cond
ition
al p
roba
bilit
y of
A g
iven
B a
s P
(A a
nd B
)/P(B
), an
d in
terp
ret i
ndep
ende
nce
of A
and
B a
s sa
ying
that
the
cond
ition
al p
roba
bilit
y of
A
giv
en B
is th
e sa
me
as th
e pr
obab
ility
of
A, a
nd th
e co
nditi
onal
pro
babi
lity
of B
gi
ven
A is
the
sam
e as
the
prob
abilit
y of
B
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M2.
S.CP
.A.3
Kno
w a
nd u
nder
stan
d th
e co
nditi
onal
pro
babi
lity
of A
giv
en
B a
s P
(A a
nd B
)/P(B
), an
d in
terp
ret
inde
pend
ence
of A
and
B a
s sa
ying
th
at th
e co
nditi
onal
pro
babi
lity
of A
gi
ven
B is
the
sam
e as
the
prob
abili
ty o
f A, a
nd th
e co
nditi
onal
pr
obab
ility
of B
giv
en A
is th
e sa
me
as th
e pr
obab
ility
of B
.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
st
anda
rd is
ass
esse
d in
this
co
urse
.
S-CP
.A.4
4.
Con
stru
ct a
nd in
terp
ret t
wo-
way
fre
quen
cy ta
bles
of d
ata
whe
n tw
o ca
tego
ries
are
asso
ciat
ed w
ith e
ach
obje
ct b
eing
cla
ssifi
ed. U
se th
e tw
o-w
ay
tabl
e as
a s
ampl
e sp
ace
to d
ecid
e if
even
ts a
re in
depe
nden
t and
to
appr
oxim
ate
cond
ition
al p
roba
bilit
ies.
Fo
r exa
mpl
e, c
olle
ct d
ata
from
a ra
ndom
sa
mpl
e of
stu
dent
s in
you
r sch
ool o
n th
eir f
avor
ite s
ubje
ct a
mon
g m
ath,
sc
ienc
e, a
nd E
nglis
h. E
stim
ate
the
prob
abilit
y th
at a
rand
omly
sel
ecte
d st
uden
t fro
m y
our s
choo
l will
favo
r sc
ienc
e gi
ven
that
the
stud
ent i
s in
tent
h gr
ade.
Do
the
sam
e fo
r oth
er s
ubje
cts
and
com
pare
the
resu
lts.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
105
★* = M
odeling Standard
S-CP.A.55.R
ecognize and explain the conceptsof conditional probability andindependence in everyday language ande veryday situations. For exam
ple,com
pare the chance of having lungcancer if you are a sm
oker with t he
c hance of being a smoker if you have
lung cancer.
There is no additional scope or clarification for this standard.
M2.S.CP.A.4 R
ecognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
S-CP.A.66.Find the conditional probability of Agiven B
as the fraction of B’s outcomes
that also belong to A, and interpret t heansw
er in terms of the m
odel.
There is no additional scope or clarification for this standard.
M2.S.CP.B.5 Find the conditional
probability of A given B as the fraction of B’s outcom
es that also belong to A and interpret the answ
er in term
s of the model.
For example, a teacher gave
two exam
s. 75 percent passed the first exam
and 25 percent passed both. W
hat percent w
ho passed the first exam also
passed the second exam?
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
S-CP.A.77.Apply the Addition R
ule, P(A or B) =P(A) + P(B) – P(A and B), and interpretthe answ
er in terms of the m
odel.
There is no additional scope or clarification for this standard.
M2.S.CP.B.6 Know
and apply the Addition R
ule, P
(A or B
) = P(A) + P
(B) –P
(A and
B), and interpret the answ
er in terms
of the model.
For example, in a m
ath class of 32 students, 14 are boys and 18 are girls. O
n a unit test 6 boys and 5 girls m
ade an A. If a student is chosen at random
from
a class, what is the
probability of choosing a girl or an A
student?
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
Nota
tion
2016
-17
Stan
dard
20
16-1
7 Sc
ope
and
Clar
ifica
tion
2017
-18
Stan
dard
20
17-1
8 Sc
ope
& Cl
arifi
catio
ns
N-Q
.A.2
2.
Def
ine
appr
opria
te q
uant
ities
for t
he
purp
ose
of d
escr
iptiv
e m
odel
ing.
Th
is s
tand
ard
will
be a
sses
sed
in
Mat
h III
by
ensu
ring
that
som
e m
odel
ing
task
s (in
volv
ing
Mat
h III
co
nten
t or s
ecur
ely
held
con
tent
fro
m p
revi
ous
grad
es a
nd c
ours
es)
requ
ire th
e st
uden
t to
crea
te a
qu
antit
y of
inte
rest
in th
e si
tuat
ion
bein
g de
scrib
ed (i
.e.,
a qu
antit
y of
in
tere
st is
not
sel
ecte
d fo
r the
st
uden
t by
the
task
). Fo
r exa
mpl
e,
in a
situ
atio
n in
volv
ing
perio
dic
phen
omen
a, th
e st
uden
t mig
ht
auto
nom
ousl
y de
cide
that
am
plitu
de is
a k
ey v
aria
ble
in a
si
tuat
ion,
and
then
cho
ose
to w
ork
with
pea
k am
plitu
de.
M3.
N.Q
.A.1
Iden
tify,
inte
rpre
t, an
d ju
stify
app
ropr
iate
qua
ntiti
es fo
r the
pu
rpos
e of
des
crip
tive
mod
elin
g.
Des
crip
tive
mod
elin
g re
fers
to
unde
rsta
ndin
g an
d in
terp
retin
g gr
aphs
; ide
ntify
ing
extra
neou
s in
form
atio
n; c
hoos
ing
appr
opria
te u
nits
; etc
. Th
ere
are
no a
sses
smen
t lim
its
for t
his
stan
dard
. The
ent
ire
stan
dard
is a
sses
sed
in th
is
cour
se.
A-SS
E.B.
2
2. U
se th
e st
ruct
ure
of a
n ex
pres
sion
to
iden
tify
way
s to
rew
rite
it. F
or e
xam
ple,
se
e x4 –
y4 a
s (x
2 )2 –
(y2 )
2 , th
us re
cogn
izin
g it
as a
diff
eren
ce o
f squ
ares
that
can
be
fact
ored
as
(x2 –
y2 )
(x2 +
y2 )
.
i) Ta
sks
are
limite
d to
pol
ynom
ial
and
ratio
nal e
xpre
ssio
ns.
ii) E
xam
ples
: see
x4 –
y4 a
s (x
2 )2 –
(y
2 )2 ,
thus
reco
gniz
ing
it as
a
diffe
renc
e of
squ
ares
that
can
be
fact
ored
as
(x2 –
y2 )
(x2 +
y2 )
. In
the
equa
tion
x2 +
2x
+ 1
+ y2 =
9, s
ee a
n op
portu
nity
to re
writ
e th
e fir
st th
ree
term
s as
(x+1
)2 , th
us r
ecog
nizi
ng
the
equa
tion
of a
circ
le w
ith ra
dius
3
and
cent
er
(-1, 0
). Se
e (x
2 + 4
)/(x2 +
3) a
s
( (x2 +
3) +
1 )/
(x2 +
3), t
hus
reco
gniz
ing
an o
ppor
tuni
ty to
writ
e it
as
1 +
1/(x
2 + 3
).
M3.
A.SS
E.A.
1 U
se th
e st
ruct
ure
of
an e
xpre
ssio
n to
iden
tify
way
s to
re
writ
e it.
For e
xam
ple,
see
2x4 +
3x2 –
5
as it
s fa
ctor
s (x
2 – 1
) and
(2x2 +
5)
; see
x4 –
y4 as
(x2 )
2 –
(y2 )
2, t
hus
reco
gniz
ing
it as
a d
iffer
ence
of
squa
res
that
can
be
fact
ored
as
(x2 –
y2 )
(x2 +
y2 )
; see
(x2 +
4)/(
x2 + 3
)
as ((
x2 + 3
) + 1
)/(x2 +
3),
thus
re
cogn
izin
g an
opp
ortu
nity
to
writ
e it
as 1
+ 1
/(x2 +
3).
Task
s ar
e lim
ited
to
poly
nom
ial,
ratio
nal,
or
expo
nent
ial e
xpre
ssio
ns.
M
3.A.
SSE.
B.2
Cho
ose
and
prod
uce
an e
quiv
alen
t for
m o
f an
expr
essi
on
to re
veal
and
exp
lain
pro
perti
es o
f th
e qu
antit
y re
pres
ente
d by
the
expr
essi
on.★
For e
xam
ple
the
expr
essi
on
1.15
t can
be
rew
ritte
n as
((1
.15)
1/12
)12t ≈
1.0
1212
t to
reve
al th
at th
e ap
prox
imat
e eq
uiva
lent
mon
thly
inte
rest
rate
107
a. Use the properties of exponents
to rewrite expressions for
exponential functions.
is 1.2% if the annual rate is
15%.
i) Tasks have a real-world
context. As described in the standard, there is an interplay betw
een the mathem
atical structure of the expression and the structure of the situation such that choosing and producing an equivalent form
of the expression reveals som
ething about the situation.
ii) Tasks are limited to
exponential expressions with
rational or real exponents
A-SSE.B.4
4. Derive the form
ula for the sum of a
finite geometric series (w
hen the com
mon ratio is not 1), and use the
formula to solve problem
s. For example,
calculate mortgage paym
ents. ★
There is no additional scope or clarification for this standard.
M3.A.SSE.B.3 R
ecognize a finite geom
etric series (when the com
mon
ratio is not 1), and know and use the
same form
ula to solve problems in
context.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A-APR.A.2 2. Know
and apply the Rem
ainder Theorem
: For a polynomial p(x) and a
number a, the rem
ainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
There is no additional scope or clarification for this standard.
M3.A.APR.A.1 Know
and apply the R
emainder Theorem
: For a polynom
ial p(x) and a num
ber 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).
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
A-APR.A.3 3. Identify zeros of polynom
ials when
suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynom
ial.
There is no additional scope or clarification for this standard.
M3.A.APR.A.2 Identify zeros of
polynomials w
hen suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynom
ial.
For example, find the zeros of
(x2 - 1)(x
2 + 1). There are no assessm
ent limits
for this standard. The entire standard is assessed in this course.
A-APR.B.4 4. Prove polynom
ial identities and use them
to describe numerical relationships.
For example, the polynom
ial identity (x
2 + y2) 2 = (x
2 – y2) 2 + (2xy) 2 can be
used to generate Pythagorean triples.
There is no additional scope or clarification for this standard.
M3.A.APR.B.3 Know
and use polynom
ial identities to describe num
erical relationships.
For example, com
pare (31)(29) = (30 + 1)(30 – 1) = 30
2 – 12
with (x + y)(x – y) = x
2 – y2.
1
A-AP
R.C.
66.
Rew
rite
sim
ple
ratio
nal e
xpre
ssio
ns in
diffe
rent
form
s; w
rite
a(x)
/b(x
) in
the
form
q(x)
+ r(
x)/b
(x),
whe
re a
(x),
b(x)
, q(x
),an
d r(x
) are
pol
ynom
ials
with
the
degr
eeof
r(x)
less
than
the
degr
ee o
f b(x
), us
ing
insp
ectio
n, lo
ng d
ivis
ion,
or,
for t
he m
ore
c om
plic
ated
exa
mpl
es, a
com
pute
ral
gebr
a sy
stem
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
A.AP
R.C.
4 R
ewrit
e ra
tiona
l ex
pres
sion
s in
diff
eren
t for
ms.
Th
ere
are
no a
sses
smen
t lim
its
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is
cour
se.
A-CE
D.A.
11.
Cre
ate
equa
tions
and
ineq
ualit
ies
inon
e va
riabl
e an
d us
e th
em to
sol
vepr
oble
ms.
Incl
ude
equa
tions
aris
ing
from
linea
r and
qua
drat
ic fu
nctio
ns, a
ndsi
mpl
e ra
tiona
l and
exp
onen
t ial
f unc
tions
.
i)Ta
sks
are
limite
d to
sim
ple
ratio
nal o
r exp
onen
tial e
quat
ions
.ii)
Task
s ha
ve a
real
-wor
ld c
onte
xt.
M3.
A.CE
D.A.
1 C
reat
e eq
uatio
ns
and
ineq
ualit
ies
in o
ne v
aria
ble
and
use
them
to s
olve
pro
blem
s.
i)Ta
sks
are
limite
d to
poly
nom
ial,
ratio
nal,
abso
lute
v alu
e, e
xpon
entia
l, or
loga
rithm
ic fu
nctio
ns.
ii)Ta
sks
have
a re
al-w
orld
cont
ext.
A-CE
D.A.
22.
Cre
ate
equa
tions
in tw
o or
mor
eva
riabl
es to
repr
esen
t rel
atio
nshi
psbe
twee
n qu
antit
ies;
gra
ph e
quat
ions
on
coor
dina
te a
xes
with
labe
ls a
nd s
cale
s.
i)Ta
sks
are
limite
d to
sim
ple
poly
nom
ial,
ratio
nal,
or e
xpon
entia
leq
uat io
ns ii
) Tas
ks h
ave
a re
al-
wor
ld c
onte
xt.
M3.
A.CE
D.A.
2 C
reat
e eq
uatio
ns in
tw
o or
mor
e va
riabl
es to
repr
esen
t re
latio
nshi
ps b
etw
een
quan
titie
s;
grap
h eq
uatio
ns w
ith tw
o va
riabl
es
on c
oord
inat
e ax
es w
ith la
bels
and
sc
ales
.
i)Ta
sks
have
a re
al-w
orld
cont
ext.
ii)Ta
sks
are
limite
d to
poly
nom
ial,
ratio
nal,
abso
lute
valu
e, e
xpon
entia
l, or
loga
rithm
ic fu
nctio
ns.
M3.
A.CE
D.A.
3 R
earra
nge
form
ulas
to
hig
hlig
ht a
qua
ntity
of i
nter
est,
usin
g th
e sa
me
reas
onin
g as
in
solv
ing
equa
tions
.
i)Ta
sks
have
a re
al-w
orld
cont
ext.
ii)Ta
sks
are
limite
d to
poly
nom
ial,
ratio
nal,
abso
lute
valu
e, e
xpon
entia
l, or
loga
rithm
ic fu
nctio
ns.
A-RE
I.A.1
1.Ex
plai
n ea
ch s
tep
in s
olvi
ng a
sim
ple
equa
tion
as fo
llow
ing
from
the
equa
lity
ofnu
mbe
rs a
sser
ted
at th
e pr
evio
us s
tep,
star
ting
from
the
assu
mpt
ion
that
the
orig
inal
equ
atio
n ha
s a
solu
tion.
Con
stru
ct a
via
ble
argu
men
t to
just
ify a
s olu
tion
met
hod.
i)Ta
sks
are
limite
d to
sim
ple
ratio
nal o
r rad
ical
equ
atio
ns.
M3.
A.RE
I.A.1
Exp
lain
eac
h st
ep in
so
lvin
g an
equ
atio
n as
follo
win
g fro
m th
e eq
ualit
y of
num
bers
as
serte
d at
the
prev
ious
ste
p,
star
ting
from
the
assu
mpt
ion
that
the
orig
inal
equ
atio
n ha
s a
solu
tion.
C
onst
ruct
a v
iabl
e ar
gum
ent t
o ju
stify
a s
olut
ion
met
hod.
Task
s ar
e lim
ited
to s
impl
e ra
tiona
l or r
adic
al e
quat
ions
A-RE
I.A.2
2.So
lve
sim
ple
ratio
nal a
nd ra
dica
leq
uatio
ns in
one
var
iabl
e, a
nd g
ive
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
A.RE
I.A.2
Sol
ve ra
tiona
l and
ra
dica
l equ
atio
ns in
one
var
iabl
e,
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd. T
he e
ntire
109
examples show
ing how extraneous
solutions may arise.
and identify extraneous solutions w
hen they exist. standard is assessed in this course.
A-REI.D.11 11. Explain w
hy the x-coordinates of the points w
here 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 approxim
ately, e.g., using technology to graph the functions, m
ake tables of values, or find successive approxim
ations. Include cases where f(x )
and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithm
ic functions. ★
i) Tasks may involve any of the
function types mentioned in the
standard.
M3.A.REI.B.3 Explain w
hy the x-coordinates of the points w
here 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 approxim
ate solutions using technology. *
Tasks may include cases
where f(x) and/or g(x) are
linear, polynomial, rational,
absolute value, exponential, or logarithm
ic functions.
F-IF.B.4 4. For a function that m
odels a relationship betw
een two quantities,
interpret key features of graphs and tables in term
s of the quantities, and sketch graphs show
ing key features given a verbal description of the relationship. K
ey features include: intercepts; intervals w
here the function is increasing, decreasing, positive, or negative; relative m
aximum
s and m
inimum
s; symm
etries; end behavior; and periodicity. ★
i) Tasks have a real-world context.
ii) Tasks may involve polynom
ial, logarithm
ic, and trigonometric
functions. The function types listed here are the sam
e as those listed in the M
ath III column for standards
F-IF.6 and F-IF.9.
M3.F.IF.A.1 For a function that
models a relationship betw
een two
quantities, interpret key features of graphs and tables in term
s of the quantities and sketch graphs show
ing key features given a verbal description of the relationship. ★
Key features include:
intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative m
aximum
s and m
inimum
s; symm
etries; and end behavior. i) Tasks have a real-w
orld context. ii) Tasks m
ay involve polynom
ial, exponential, and logarithm
ic functions.
F-IF.B.6 6. C
alculate and interpret the average rate of change of a function (presented sym
bolically or as a table) over a specified interval. Estim
ate the rate of change from
a graph. ★
i) Tasks have a real-world context.
ii) Tasks may involve polynom
ial, logarithmic,
and trigonometric functions. The function types
listed here are the same as those listed in the
Math III colum
n for standards F-IF.4 and F-IF.9.
M3.F.IF.A.2 C
alculate and interpret the average rate of change of a function (presented sym
bolically or as a table) over a specified interval. Estim
ate the rate of change from a
graph. *
i) Tasks have a real-world
context. ii) Tasks m
ay involve polynom
ial, exponential, and logarithm
ic functions.
F-IF.C.7 7. G
raph functions expressed sym
bolically and show key features of
the graph, by hand in simple cases and
using technology for more com
plicated cases. ★
For F-IF.7e: i) Tasks are lim
ited to logarithmic and
trigonometric functions.
M3.F.IF.B.3 G
raph functions expressed sym
bolically and show
key features of the graph, by hand and using technology. ★
* a.
Graph linear and quadratic
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
c. G
raph
pol
ynom
ial f
unct
ions
, id
entif
ying
zer
os w
hen
suita
ble
fact
oriz
atio
ns a
re a
vaila
ble,
and
sho
win
g en
d be
havi
or.
e. G
raph
exp
onen
tial a
nd lo
garit
hmic
fu
nctio
ns, s
how
ing
inte
rcep
ts a
nd e
nd
beha
vior
, and
trig
onom
etric
func
tions
, sh
owin
g pe
riod,
mid
line,
and
am
plitu
de.
func
tions
and
sho
w
inte
rcep
ts, m
axim
a, a
nd
min
ima.
b.
G
raph
squ
are
root
, cub
e ro
ot, a
nd p
iece
wis
e-de
fined
fu
nctio
ns, i
nclu
ding
ste
p fu
nctio
ns a
nd a
bsol
ute
valu
e fu
nctio
ns.
c.
Gra
ph p
olyn
omia
l fun
ctio
ns,
iden
tifyi
ng z
eros
whe
n su
itabl
e fa
ctor
izat
ions
are
av
aila
ble
and
show
ing
end
beha
vior
. d.
G
raph
exp
onen
tial a
nd
loga
rithm
ic fu
nctio
ns,
show
ing
inte
rcep
ts a
nd e
nd
beha
vior
.
F-IF
.C.9
9.
Com
pare
pro
perti
es o
f tw
o fu
nctio
ns
each
repr
esen
ted
in a
diff
eren
t way
(a
lgeb
raic
ally
, gra
phic
ally
, num
eric
ally
in
tabl
es, o
r by
verb
al d
escr
iptio
ns).
For
exam
ple,
giv
en a
gra
ph o
f one
qua
drat
ic
func
tion
and
an a
lgeb
raic
exp
ress
ion
for
anot
her,
say
whi
ch h
as th
e la
rger
m
axim
um.
i) Ta
sks
have
a re
al-w
orld
con
text
. ii)
Tas
ks m
ay in
volv
e po
lyno
mia
l, lo
garit
hmic
, and
trig
onom
etric
fu
nctio
ns.
The
func
tion
type
s lis
ted
here
are
th
e sa
me
as th
ose
liste
d in
the
Mat
h III
col
umn
for s
tand
ards
F-
IF.4
and
F-IF
.6.
M3.
F.IF
.B.4
Com
pare
pro
perti
es o
f tw
o fu
nctio
ns e
ach
repr
esen
ted
in a
di
ffere
nt w
ay (a
lgeb
raic
ally
, gr
aphi
cally
, num
eric
ally
in ta
bles
, or
by v
erba
l des
crip
tions
).
Task
s m
ay in
volv
e po
lyno
mia
l, ex
pone
ntia
l, an
d lo
garit
hmic
fu
nctio
ns.
F-BF
.B.3
3.
Iden
tify
the
effe
ct o
n th
e gr
aph
of
repl
acin
g f(x
) by
f(x) +
k, k
f(x)
, f(k
x), a
nd
f(x +
k) f
or s
peci
fic v
alue
s of
k (b
oth
posi
tive
and
nega
tive)
; fin
d th
e va
lue
of k
gi
ven
the
grap
hs. E
xper
imen
t with
cas
es
and
illus
trate
an
expl
anat
ion
of th
e ef
fect
s on
the
grap
h us
ing
tech
nolo
gy.
Incl
ude
reco
gniz
ing
even
and
odd
fu
nctio
ns fr
om th
eir g
raph
s an
d al
gebr
aic
expr
essi
ons
for t
hem
.
i) Ta
sks
are
limite
d to
exp
onen
tial,
poly
nom
ial,
loga
rithm
ic, a
nd
trigo
nom
etric
func
tions
. ii)
Tas
ks m
ay in
volv
e re
cogn
izin
g ev
en a
nd o
dd fu
nctio
ns. T
he
func
tion
type
s lis
ted
in n
ote
(i) a
re
the
sam
e as
thos
e lis
ted
in th
e M
ath
III c
olum
n fo
r sta
ndar
ds F
-IF
.4, F
-IF.6
and
F-IF
.9.
M3.
F.BF
.A.1
Iden
tify
the
effe
ct o
n th
e gr
aph
of re
plac
ing
f(x) b
y f(x
) + k
, kf
(x),
f(kx)
, and
f(x
+ k)
for s
peci
fic
valu
es o
f k (b
oth
posi
tive
and
nega
tive)
; fin
d th
e va
lue
of k
giv
en
the
grap
hs. E
xper
imen
t with
cas
es
and
illus
trate
an
expl
anat
ion
of th
e ef
fect
s on
the
grap
h us
ing
tech
nolo
gy.
i) Ta
sks
may
invo
lve
poly
nom
ial,
expo
nent
ial,
and
loga
rithm
ic fu
nctio
ns.
ii) T
asks
may
invo
lve
reco
gniz
ing
even
and
odd
fu
nctio
ns.
F-BF
.B.4
4.
Fin
d in
vers
e fu
nctio
ns.
a. S
olve
an
equa
tion
of th
e fo
rm f(
x) =
c
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n in
form
atio
n fo
r thi
s M
3.F.
BF.A
.2 F
ind
inve
rse
func
tions
. a.
Fin
d th
e in
vers
e of
a fu
nctio
n Th
ere
are
no a
sses
smen
t lim
its
for t
his
stan
dard
. Th
e en
tire
111
for a simple function f that has an
inverse and write an expression for the
inverse. For example,
f(x) =2x3 or f(x) = (x+1)/(x–1) for x ≠ 1.
standard. w
hen the given function is one-to-one.
standard is assessed in this course.
M3.F.LE.A.1 O
bserve using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (m
ore generally) as a polynom
ial function.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
F-LE.B.4 4. For exponential m
odels, express as a logarithm
the solution to abct = d w
here a, c, and d are num
bers and the base b is 2, 10, or e; evaluate the logarithm
using technology.
There is no additional scope or clarification for this standard.
M3.F.LE.A.2 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.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
F-TF.A.1 1. U
nderstand radian measure of an
angle as the length of the arc on the unit circle subtended by the angle.
There is no additional scope or clarification for this standard.
M3.F.TF.A.1 U
nderstand and use radian m
easure of an angle. a. U
nderstand radian measure of an
angle as the length of the arc on the unit circle subtended by the angle. b. U
se the unit circle to find sin θ, cos θ, and tan θ w
hen θ is a com
monly recognized angle
between 0 and 2π
.
Com
monly recognized angles
include all multiples of nπ
/6 and nπ /4, w
here n is an integer. There are no assessm
ent limits
for this standard. The entire standard is assessed in this course.
F-TF.A.2 2. Explain how
the unit circle in the coordinate plane enables the extension of trigonom
etric functions to all real num
bers, interpreted as radian m
easures of angles traversed counterclockw
ise around the unit circle.
There is no additional scope or clarification for this standard.
M3.F.TF.A.2 Explain how
the unit circle in the coordinate plane enables the extension of trigonom
etric functions to all real num
bers, interpreted as radian m
easures of angles traversed counterclockw
ise around the unit circle.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
F-TF.B.5 5. C
hoose trigonometric functions to
model periodic phenom
ena with
specified amplitude, frequency, and
midline. ★
There is no additional scope or clarification for this standard.
F-TF
.C.8
8.
Pro
ve th
e Py
thag
orea
n id
entit
y si
n2 (θ)
+
cos2 (
θ) =
1 a
nd u
se it
to fi
nd s
in(θ
), co
s(θ)
, or t
an(θ
) giv
en s
in(θ
), co
s(θ)
, or
tan(
θ) a
nd th
e qu
adra
nt o
f the
ang
le.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
F.TF
.B.3
Use
trig
onom
etric
id
entit
ies
to fi
nd v
alue
s of
trig
fu
nctio
ns.
a. G
iven
a p
oint
on
a ci
rcle
cen
tere
d at
the
orig
in, r
ecog
nize
and
use
the
right
tria
ngle
ratio
def
initi
ons
of s
in
θ, c
os θ
, and
tan
θ to
eva
luat
e th
e tri
gono
met
ric fu
nctio
ns.
b. G
iven
the
quad
rant
of t
he a
ngle
, us
e th
e id
entit
y si
n2 θ +
cos
2 θ =
1 to
fin
d si
n θ
give
n co
s θ,
or v
ice
vers
a.
Com
mon
ly re
cogn
ized
ang
les
incl
ude
all m
ultip
les
of n
π /6
an
d
nπ /4
, whe
re n
is a
n in
tege
r.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
co
urse
.
F-CO
.D.1
2 12
. Mak
e fo
rmal
geo
met
ric c
onst
ruct
ions
w
ith a
var
iety
of t
ools
and
met
hods
(c
ompa
ss a
nd s
traig
hted
ge, s
tring
, re
flect
ive
devi
ces,
pap
er fo
ldin
g,
dyna
mic
geo
met
ric s
oftw
are,
etc
.).
Cop
ying
a s
egm
ent;
copy
ing
an a
ngle
; bi
sect
ing
a se
gmen
t; bi
sect
ing
an a
ngle
; co
nstru
ctin
g pe
rpen
dicu
lar l
ines
, inc
ludi
ng th
e pe
rpen
dicu
lar b
isec
tor o
f a li
ne s
egm
ent;
and
cons
truct
ing
a lin
e pa
ralle
l to
a gi
ven
line
thro
ugh
a po
int n
ot o
n th
e lin
e.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
G.C
O.A
.1 M
ake
form
al
geom
etric
con
stru
ctio
ns w
ith a
va
riety
of t
ools
and
met
hods
(c
ompa
ss a
nd s
traig
hted
ge, s
tring
, re
flect
ive
devi
ces,
pap
er fo
ldin
g,
dyna
mic
geo
met
ric s
oftw
are,
etc
.).
Con
stru
ctio
ns in
clud
e bu
t are
no
t lim
ited
to: c
opyi
ng a
se
gmen
t; co
pyin
g an
ang
le;
bise
ctin
g a
segm
ent;
bise
ctin
g an
ang
le; c
onst
ruct
ing
perp
endi
cula
r lin
es, i
nclu
ding
th
e pe
rpen
dicu
lar b
isec
tor o
f a
line
segm
ent;
cons
truct
ing
a lin
e pa
ralle
l to
a gi
ven
line
thro
ugh
a po
int n
ot o
n th
e lin
e,
and
cons
truct
ing
the
follo
win
g ob
ject
s in
scrib
ed in
a c
ircle
: an
equi
late
ral t
riang
le, s
quar
e,
and
a re
gula
r hex
agon
.
F-CO
.D.1
3 13
. Con
stru
ct a
n eq
uila
tera
l tria
ngle
, a
squa
re, a
nd a
regu
lar h
exag
on in
scrib
ed
in a
circ
le.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
G-C
.A.1
1.
Pro
ve th
at a
ll ci
rcle
s ar
e si
mila
r. Th
ere
is n
o ad
ditio
nal s
cope
or
clar
ifica
tion
for t
his
stan
dard
. M
3.G
.C.A
.1 R
ecog
nize
that
all
circ
les
are
sim
ilar.
Th
ere
are
no a
sses
smen
t lim
its
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is
cour
se.
G-C
.A.2
2.
Iden
tify
and
desc
ribe
rela
tions
hips
am
ong
insc
ribed
ang
les,
radi
i, an
d ch
ords
. Inc
lude
the
rela
tions
hip
betw
een
cent
ral,
insc
ribed
, and
circ
umsc
ribed
an
gles
; ins
crib
ed a
ngle
s on
a d
iam
eter
ar
e rig
ht a
ngle
s; th
e ra
dius
of a
circ
le is
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
G.C
.A.2
Iden
tify
and
desc
ribe
rela
tions
hips
am
ong
insc
ribed
an
gles
, rad
ii, a
nd c
hord
s.
Incl
ude
the
rela
tions
hip
betw
een
cent
ral,
insc
ribed
, and
ci
rcum
scrib
ed a
ngle
s; in
scrib
ed
angl
es o
n a
diam
eter
are
righ
t an
gles
; the
radi
us o
f a c
ircle
is
perp
endi
cula
r to
the
tang
ent
113
perpendicular to the tangent where the
radius intersects the circle. w
here the radius intersects the circle, and properties of angles for a qu adrilateral inscribed in a circle.
G-C.A.3
3.Construct the inscribed and
circumscribed circles of a triangle, and
prove properties of angles for aquadrilateral inscribed in a circle.
There is no additional scope or clarification for this standard.
M3.G
.C.A.3 Construct the incenter
and circumcenter of a triangle and
use their properties to solve problem
s in context.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G-C.B.5
5.Derive using sim
ilarity the fact that thelength of the arc intercepted by an angleis proportional to the radius, and defi net he radian m
easure of the angle as theconstant of proportionality; derive theform
ula for the area of a sector.
There is no additional scope or clarification for this standard.
M3.G
.C.B.4 Find the area of a sector of a circle in a real w
orld context.
For example, use proportional
relationships and angles m
easured in degrees or radians.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G-G
PE.A.11.D
erive the equation of a circle of givencenter and radius using the PythagoreanTheorem
; complete the square to fi nd
t he center and radius of a circle given byan equation.
There is no additional scope or clarification for this standard.
M3.G
.GPE.A.1 W
rite the equation of a circle of given center and radius using the Pythagorean Theorem
.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G-G
PE.A.22.D
erive the equation of a parabolagiven a focus and directrix.
There is no additional scope or clarification for this standard.
G-G
PE.B.44.U
se coordinates to prove simple
geometric theorem
s algebraically. Forexam
ple, prove or disprove that a figuredefined by four given points in thecoordinate plane is a rectangle; prove ordisprove that the point (1, √3) lies on t hec ircle centered at the origin andcontaining the point (0, 2).
There is no additional scope or clarification for this standard.
M3.G
.GPE.B.2 U
se coordinates to prove sim
ple geometric theorem
s 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).
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
G-G
PE.B
.55.
Prov
e th
e sl
ope
crite
ria fo
r par
alle
lan
d pe
rpen
dicu
lar l
ines
and
use
them
toso
lve
geom
etric
pro
blem
s (e
.g.,
find
the
equa
tion
of a
line
par
alle
l or
perp
endi
cula
r to
a gi
ven
line
that
pas
ses
thro
ugh
a gi
ven
poin
t).
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
G.G
PE.B
.3 P
rove
the
slop
e cr
iteria
for p
aral
lel a
nd p
erpe
ndic
ular
lin
es a
nd u
se th
em to
sol
ve
geom
etric
pro
blem
s.
For e
xam
ple,
find
the
equa
tion
of a
line
par
alle
l or
perp
endi
cula
r to
a gi
ven
line
that
pas
ses
thro
ugh
a gi
ven
poin
t.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
co
urse
.
G-G
PE.B
.66.
Find
the
poin
t on
a di
rect
ed li
nese
gmen
t bet
wee
n tw
o gi
ven
poin
ts th
atpa
rtitio
ns th
e se
gmen
t in
a gi
ven
ratio
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
G.G
PE.B
.4 F
ind
the
poin
t on
a di
rect
ed li
ne s
egm
ent b
etw
een
two
give
n po
ints
that
par
titio
ns th
e se
gmen
t in
a gi
ven
ratio
.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
co
urse
.
G-G
PE.B
.77.
Use
coo
rdin
ates
to c
ompu
tepe
rimet
ers
of p
olyg
ons
and
area
s of
trian
gles
and
rect
angl
es, e
.g.,
usin
g th
edi
stan
ce fo
rmul
a.★
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
G.G
PE.B
.5 U
se c
oord
inat
es to
co
mpu
te p
erim
eter
s of
pol
ygon
s an
d ar
eas
of tr
iang
les
and
rect
angl
es.*
For e
xam
ple,
use
the
dist
ance
fo
rmul
a.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
co
urse
.
G-G
MD.
A.4
4.Id
entif
y th
e sh
apes
of t
wo-
dim
ensi
onal
cros
s-se
ctio
ns o
f thr
ee-d
imen
sion
alob
ject
s, a
nd id
entif
y th
ree-
dim
ensi
onal
obje
cts
gene
rate
d by
rota
tions
of t
wo-
dim
ensi
onal
obj
ects
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
G-M
G.A
.11.
Use
geo
met
ric s
hape
s, th
eir
mea
sure
s, a
nd th
eir p
rope
rties
tode
s crib
e ob
ject
s (e
.g.,
mod
elin
g a
tree
trunk
or a
hum
an to
rso
as a
cyl
inde
r).★
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
G.M
G.A
.1 U
se g
eom
etric
sh
apes
, the
ir m
easu
res,
and
thei
r pr
oper
ties
to d
escr
ibe
obje
cts.
★
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
co
urse
.
G-M
G.A
.22.
Appl
y co
ncep
ts o
f den
sity
bas
ed o
nar
ea a
nd v
olum
e in
mod
elin
g si
tuat
ions
(e.g
., pe
rson
s pe
r squ
are
mile
, BTU
s pe
rcu
bic
foot
).★
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
G-M
G.A
.33.
Appl
y ge
omet
ric m
etho
ds to
sol
vede
sign
pro
blem
s (e
.g.,
desi
gnin
g an
obje
ct o
r stru
ctur
e to
sat
isfy
phy
sica
lco
nstra
ints
or m
inim
ize
cost
; wor
king
with
typo
grap
hic
grid
sys
tem
s ba
sed
on
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
G.M
G.A
.2 A
pply
geo
met
ric
met
hods
to s
olve
real
wor
ld
prob
lem
s.★
Geo
met
ric m
etho
ds m
ay
incl
ude
but a
re n
ot li
mite
d to
us
ing
geom
etric
sha
pes,
the
prob
abilit
y of
a s
hade
d re
gion
, de
nsity
, and
des
ign
prob
lem
s.
115
ratios). ★ There are no assessm
ent limits
for this standard. The entire standard is assessed in this course.
S-ID.A.44.U
se the mean and standard deviation
of a data set to fit it to a normal
distribution and to estimate populati on
per centages. 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.
There is no additional scope or clarification for this standard.
M3.S.ID.A.1 U
se the mean and
standard deviation of a data set to fit it to a norm
al distribution and to estim
ate population percentages using the Em
pirical Rule.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
S-ID.B.66.R
epresent data on two quantitative
variables on a scatter plot, and describehow
the variables are related.a.Fit a function to the data; usefunctions fitted to data to solve problem
sin the context of the data. U
se givenfunctions or choose a function suggestedby the context. E
mphasize linear,
quadratic, and exponential models.
b.Informally assess the fit of a function
by plotting and analyzing residuals.
For S-ID.6a:
i)Tasks have a real-world context.
ii)Tasks are limited to exponential
functions with dom
ains not in thei ntegers and trigonom
etricfunctions
M3.S.ID.B.2 R
epresent data on two
quantitative variables on a scatter plot, and describe how
the variables are related. a.Fit a function to the data; usef unctions fitted to data to solveproblem
s in the context of the data.U
se given functions or choos e afunction suggested by the context.b.Fit a linear function for a scatterplot that suggests a linearassociation.
Use given functions or choose
a function suggested by the context.
i)Tasks have a real-wor ld
c ontext.
ii)T asks are limited to linear,
quadratic, and exponentialfunctions w
ith domains not in
t he integers.
S-IC.A.11.U
nderstand statistics as a process form
aking inferences about populationparam
eters based on a random sam
pl ef rom
that population.
There is no additional scope or clarification for this standard.
M3.S.IC.A.1 U
nderstand statistics as a process for m
aking inferences about population param
eters based on a random
sample from
that population.
There are no assessment lim
its for this standard. The entire standard is assessed in this course.
S-IC.A.22.D
ecide if a specified model is
consistent with results from
a given data-generating process, e.g., usi ngs im
ulation. For example, a m
odel says aspinning coin falls heads up w
ithprobability 0.5. W
ould a result of 5 tailsin a row
cause you to question them
odel?
There is no additional scope or clarification for this standard.
M3.S.IC.A.2 D
ecide if a specified m
odel is consistent with results from
a given data-generating process, e.g., using sim
ulation.
For example, a m
odel says a spinning coin falls heads up w
ith probability 0.5. Would a
result of 10 heads in a row
cause you to question the m
odel? There are no assessm
ent limits
for this standard. The entire standard is assessed in this course.
★* =
Mod
elin
g St
anda
rd
S-IC
.B.3
3.R
ecog
nize
the
purp
oses
of a
nddi
ffere
nces
am
ong
sam
ple
surv
eys,
expe
rimen
ts, a
nd o
bser
vatio
nal s
tudi
es;
expl
ain
how
rand
omiz
atio
n re
late
s to
each
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
S.IC
.B.3
Rec
ogni
ze th
e pu
rpos
es o
f and
diff
eren
ces
amon
g sa
mpl
e su
rvey
s, e
xper
imen
ts, a
nd
obse
rvat
iona
l stu
dies
; exp
lain
how
ra
ndom
izat
ion
rela
tes
to e
ach.
For e
xam
ple,
in a
giv
en
situ
atio
n, is
it m
ore
appr
opria
te
to u
se a
sam
ple
surv
ey, a
n ex
perim
ent,
or a
n ob
serv
atio
nal s
tudy
? E
xpla
in
how
rand
omiz
atio
n af
fect
s th
e bi
as in
a s
tudy
. Th
ere
are
no a
sses
smen
t lim
its
for t
his
stan
dard
. Th
e en
tire
stan
dard
is a
sses
sed
in th
is
cour
se.
S-IC
.B.4
4.U
se d
ata
from
a s
ampl
e su
rvey
toes
timat
e a
popu
latio
n m
ean
orpr
opor
tion;
dev
elop
a m
argi
n of
erro
rth
roug
h th
e us
e of
sim
ulat
ion
mod
els
for
rand
om s
ampl
ing.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
M3.
S.IC
.B.4
Use
dat
a fro
m a
sam
ple
surv
ey to
est
imat
e a
popu
latio
n m
ean
or p
ropo
rtion
; use
a g
iven
m
argi
n of
erro
r to
solv
e a
prob
lem
in
cont
ext.
Ther
e ar
e no
ass
essm
ent l
imits
fo
r thi
s st
anda
rd.
The
entir
e st
anda
rd is
ass
esse
d in
this
co
urse
.
S-IC
.B.5
5.U
se d
ata
from
a ra
ndom
ized
expe
rimen
t to
com
pare
two
treat
men
ts;
use
sim
ulat
ions
to d
ecid
e if
diffe
renc
esbe
twee
n pa
ram
eter
s ar
e si
gnifi
cant
.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
S-IC
.B.6
6.Ev
alua
te re
ports
bas
ed o
n da
ta.
Ther
e is
no
addi
tiona
l sco
pe o
r cl
arifi
catio
n fo
r thi
s st
anda
rd.
117
Standards Comparison ChartStandard
CodingDropped
from CourseAdded to
Course No change
118
Revised
Revisions to the Math Standards
Standards Comparison Activity
1. If you had to summarize the revisions to these selected standards in twenty words or less, what would you say?
Notes:
Small Group Consensus:
Whole Group Consensus:
119
Notes:
Appointment with Peers
Please meet with your first partner to discuss the following:
• How will these changes impact your classroom?
• What are your takeaways from modules 1–3?
• How does this align to your observation rubric?
120
Module 3 Review
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
• The instructional expectations remain the same and are still the focus of the standards.
• The revised standards represent a stronger foundation that will support the progression of rigorous standards throughout the grade levels.
• The revised standards improve connections:
– within a single grade level, and
– between multiple grade levels.
121
Strong StandardsHigh
Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 2: Developing a Deeper UnderstandingModule 4: Diving Into 9–12 Math
123
TAB PAGE
Goals• Concisely describe a course based on its introduction.
emphasis of the standard—its intent and purpose.
• Use the KUD approach to guide planning, instruction, and assessment.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standardsarethebricksthatshouldbemasterfullylaidthroughqualityinstructiontoensurethatallstudentsreachtheexpectationofthestandards.
Instructional Shifts
Theinstructionalshiftsareanessentialcomponentofthestandardsandprovideguidanceforhowthestandardsshouldbetaughtandimplemented.
Aligned Materials and Assessments
Educatorsplayakeyroleinensuringthatourstandards,classroominstructionalmaterials,andassessmentsarealigned.
125
• Develop a means for deconstructing standards to determine the mathematical
Closer Look
Take a few minutes to read the overview page for your grade level and think about how this relates to the overarching revisions we have just seen.
Notes:
Now summarize your course in 140 characters. Write your tweet to inform others regarding what is included in your grade.
My Tweet:
126
“With my ears to the ground, listening to my students, my eyes are focused on the mathematical horizon.”
—Ball, 1993
Intent and Purpose of the Standards
Analyzing Standards
A1.F.IF.C.7 (Note also: A2.F.IF.B.4 and M2.F.IF.B.5)
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. 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.
Notes:
127
We are going to look closely at A1.F.IF.C.7.
Know (facts, vocabulary)
Understand(concepts,
generalizations)
Do(verbs, skills)
Essential Questions:
Instruction & Assessment (What does the math look like?)
128
• What is it that the standard wants the student to know, understand, and do?
• KUD – helps to maintain focus in differentiated instruction
– Know: facts, vocabulary, properties, procedures, etc.
– Understand: concepts, ideas, etc.
– Do: tasks, approaches, assessment problems, etc.
• The two go together: What is the intent and purpose of the standard and how do I put this into instructional form?
From Standard to Instruction: KUD
Know, Understand, and Do
Know
Understand
Do
129
You Try One!
Know (facts, vocabulary)
Understand(concepts,
generalizations)
Do(verbs, skills)
Essential Questions:
Instruction & Assessment (What does the math look like?)
130
Module 4 Review
High Expectations
We have a continued goal to prepare students to be college and career ready.
• Concisely describe a course based on its introduction.
• Develop a means for deconstructing standards to determine the mathematical emphasis, intent, and purpose of the standard.
• Use the KUD approach to guide planning, instruction, and assessment.
131
Strong StandardsHigh
Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 3: Instructional ShiftsModule 5: Revisiting the SMP’s and Instructional Expectations
133
TAB PAGE
Goals
• Revisit the concepts of focus, coherence, and rigor and how they play out in instruction.
• Discuss the purpose and place of the content and practice standards.
• Share instructional strategies related to the Standards for Mathematical Practice.
• Discuss research on the influence of mindsets in the math classroom.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
Instructional Shifts
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Aligned Materials and Assessments
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
135
• Explore students’ mathematical mindsets.
Why Standards for Mathematical Practice?
“Beginning to experiment with small changes to one’s teaching practice and collaborating with colleagues can help move students toward the vision of mathematical proficiency described in the Standards for Mathematical Practice”
—Mateas, 2016
• Tell us what students should know and be able to do
• So, what should students know and do?
• Content Standards
• Standards for Mathematical Practice
• Literacy Skills
• Knowing that these are what students need to learn, teachers determine howto teach these.
136
Mindset
Notes:
137
• The TN Academic Standards for Mathematics may seem challenging for students whose mindsets have been fixed by their past experiences in mathematics classrooms.
• As teachers, we are best positioned to influence students’ mathematical mindsets through our actions and practices in the mathematics classroom.
_____________________________ -Intelligence is a fixed trait. You cannot change it.
_____________________________ -You can grow your intelligence through effort.
Notes:
Why Address Mindsets?“If there’s a threat of being wrong every time I raise my hand, and being wrong is a bad thing, then very quickly I decide math isn’t for me, I don’t like this, I’m not a smart person.”
—Noah Heller, Harvard Graduate School of Education, 2016
Everyone can learn math to
the highest levels
Mistakes are valuable
Questions are important
Math is about creativity and making sense
Math is about connections
and communicating
Math class is about learning not preforming
Depth is more important than
speed
138
Instructional Expectations
Focus1. In your grade-level groups, discuss ways you could respond if someone asks you
the following question, “Why focus? There’s so much math that students could be learning. Why limit them?”
2. Review the table below and answer the questions, “Which two of the following represent areas of major focus for the indicated grade?”
8 Standard form of a linear equation.
Define, evaluate, and compare functions.
Understand and apply the Pythagorean Theorem.
Alg. 1 Zeros of polynomials Linear and quadratic functions.
Creating equations to model situations.
Alg. 2 Exponential andlogarithmic functions. Polar coordinates Using fractions to
model situations.
139
Instructional Shifts
Coherence
In the space below, copy all of the standards for your assigned domain and note how coherence is evident in the vertical progression of these standards.
Grade Standard Summary of the Standard (If the standard has sub-parts,summarize each sub-part.)
140
Instructional Shifts
Rigor 1. Make a true statement: Rigor = _______________ + _______________ + _______________
2. In your groups, discuss ways to respond to one of the following comments: “These standards are expecting that we just teach rote memorization. Seems like a step backwards to me.” Or “I’m not going to spend time on fluency—it should just be a natural outcome of conceptual understanding.”
3. The shift towards rigor is required by the standards. Find and copy in the space below standards which specifically set expectations for each component of rigor.
141
EvidenceStandard
Instructional Shifts
The instructional expectations are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Module 5 Review
• We connected the instructional shifts to the standards and our classroom practices.
• We explored students’ mathematical mindsets.
• We shared instructional strategies related to the Standards for Mathematical Practice.
142
What do the instructional shifts look like in the classroom?
Strong StandardsHigh
Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 3: Instructional ShiftsModule 6: Literacy Skills for Mathematical Proficiency
143
TAB PAGE
Goal
• Develop a better understanding of the Literacy Skills for Mathematical Proficiency.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
Instructional Shifts
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Aligned Materials and Assessments
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
144
Reflect on ways literacy skills are already present in your mathematics classroom.
Literacy in your Math Classroom
Literacy Skills for Math ProficiencyCommunication in mathematics requires literacy skills in reading, vocabulary, speaking, listening, and writing.
145
Reading
Vocabulary
Speaking & Listening
Writing
Literacy Skills for Math ProficiencyCategorize the strategies you listed and discussed with your table partners in the chart below.
146
Literacy Skills for Mathematical Proficiency
1. Read and annotate your assigned section from pages 13–14 of the TN Math Standards. Work with your group to present this information to your colleagues.
2. Use the chart below to take notes and highlight the main ideas of each section.
Reading
Vocabulary
Speaking & Listening
Writing
147
Strategies for Incorporating LiteracyText Features
Highlight key symbols
Color code steps or circle action steps
Place a box around key terms in vocabulary
Notes:
148
Brandon and Allison participate in an annual community 5k. Brandon can run at a rate
of 3 miles in 24 minutes. Allison can run at a rate of 1 mile in 9 minutes. Who has the
faster rate? Who finished the race first?
Let’s Try a Problem
Strategies for Incorporating LiteracyGraphic Organizers
Graphs
Tables
Four Corner’s and a Diamond
Frayer’s Model
Four Square Graphic Organizer
Semantic Grid Analysis
Originally from Teaching Children Mathematics, © November 2009, Mathematical graphic organizers, p. 222.
May be adapted for personal use with students.
in
What do you
need to find?
Four-Corners-and-a-Diamond Math Graphic Organizer
What do you already know? Brainstorm ways to solve this problem.
Try it here. Explanations you need to include in your
extended-response write up
149
Notes:
Four Stages of Word Knowledge
Mathematics Vocabulary
150
2. Students have seen/heard the word but do not know the definition.
1. Students have never encountered the word before.
3. Students know the word but rely on context to define it.
4. Students know the word and can use it comfortably.
Mathematical Vocabulary• Student achievement is dependent upon students’ reading comprehension and
content area learning.
• Math vocabulary is decontextualized because they are not in everyday conversations.
• Terms in math can have multiple meanings– i.e. table, origin, and leg.
• Mathematical terms can have specific meanings – i.e. average, reflection.
• Students need to develop a conceptual meaning and read use the words accurately.
Frayer’s Model
151
Module 6 Review
Instructional Shifts
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
• Literacy skills in the math classroom will support students’ understanding of the content standards.
• When students can read, write, and speak about math ideas, connections are made between concepts.
152
Making Connections
153
Notes:
Appointment with Peers
Please meet with your second partner to discuss the following:
• What are your key takeaways from today?
• How does the align to your observation rubric?
154
Strong Standards
High Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 4: Assessment and MaterialsModule 7: Connecting Standards and Assessment
155
TAB PAGE
Goals
• Discuss the role assessment plays in the integrated system of learning.
• Discuss the cycle of assessment.
• Discuss the areas of focus for standards-aligned assessments.
• Review and write mathematics assessment items.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
Instructional Shifts
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Aligned Materials and Assessments
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
156
Connecting Standards and Assessment
Assessment
Curriculum
Instruction
Teacher Development
Standards
Assessment is_____________________________________________________________________________.
Considering this definition of assessment, what are educators “making a judgement about” when assessing students?
157
The Cycle of Assessment
“The good news is that research has shown for years that consistently applying principles of assessment for learning has yielded remarkable, if not unprecedented, gains in student achievement, especially for low achievers.”
—Black & Wiliam, 1998
Teach
Assess
Analyze
Action
158
The Cycle of Assessment
Areas of Focus1. Intent of the Assessment
• Summative
• Formative
2. Content and structure of Assessments
3. Analysis of Assessments
Teach
Assess
Analyze
Action
Standards Aligned Assessment
159
“Benchmark assessments, either purchased by the district or from commercial vendors or developed locally, are generally meant to measure progress toward state or district content standards and to predict performance on large-scale summative tests. A common misconception is that this level of assessment is automatically formative.”
—Stephen and Jan Chappuis 2012
Intent of Assessments
Areas of Focus1. Intent of the Assessment
• Summative• Formative
2. Content and Structure of Assessments
3. Analysis of Assessments
How are the results used?
Formative Summative
160
Intent of Assessments
Areas of Focus1. Intent of the Assessment
• Summative
• Formative
2. Content and Structure of Assessments
3. Analysis of Assessments
Things to think about…Universal Design Principles:
• No barriers
• Accessible for all students
• Upholds the expectations of our state standards
Notes:
161
Developing a Classroom Assessment
Identify targeted
standards
What essential understandings do I want my students to display mastery
of now?
Deconstruct standards
What types of questions
should I ask?
Will this generate the
data that I really need?
Identify essential understandings
Notes:
162
Inventory for a Classroom Assessment
Purpose of Assessment
Formative
What items do I have? Assess Items
What items do I still need? Create Items
Summative
What items do I have? Assess Items
What items do I still need? Create Items
Notes:
163
Item ReviewStandard:4.OA.A.3: Solve multi-step contextual problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Item 2: Samantha bought stickers.
• She bought 6 packs of stickers.• Each pack has 12 stickers.• She got 8 more stickers from a
friend.
How many stickers does Samantha have in all?A. 26B. 64C. 72D. 80
Item 1: Samantha bought stickers.
• She bought 6 packs of stickers.• Each pack has 12 stickers.• She got 8 more stickers from a
friend.
How many stickers does Samantha have in all?A. 76B. 78C. 80D. 82
Which item is better?
Notes:
164
Item Review
Assessment Terminology
Item Type
Selected response
Open response
Verbal
Extended writing
Item Components
Stimulus
Stem
Key
Distractor
165
Rationale
Examining Items: Formative vs. Summative
• What is the question actually asking?
• Is the question aligned to the depth of the standard?
• Are the answers precise?
• Is the wording grade appropriate?
• Is the question aligned to the standard?
• Do the distractors give insight into student thinking?
• Is the entire standard assessed?
• Is the question precise?
• Is there a better way to assess the standard?
166
You will be looking at five items. Decide if you would keep them, revise them in some way, or throw the item out all together. Look first at the items independently. Then you may work with a partner to complete the activity.
Item Assessment Activity
Item #2
G.SRT.B.4 (M2.G.SRT.B.4)
Know and write the equation of a circle of given center and radius using the Pythagorean Theorem.
Item #1
M3.G.GPE.A.1 (G.GPE.A.1)
Prove theorems about similar triangles.
Right Triangle ABC has side lengths 5, 12 and 13. If Triangle DEF is similar to ABC and has one side length 60, what are the possible missing side lengths of DEF?
A. 10 and 24B. 25 and 65C. 30 and 78D. 36 and 96E. 100 and 125F. 144 and 156
The equation for a given circle is 2x2 + 2y2 – 8x – 12y + 8 = 0.
What is the radius of the circle?
A. 2B. 3C. 4D. 12
167
For each of the provided formative assessment items, think about the things we just discussed. Would you include it on a formative assessment when paired with the provided standard?
Item Assessment Activity
Item #3
A2.A.REI.D.6 (M1.A.REI.C.4)
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the approximate solutions using technology.
!(")=|"|− 2 and #(")= −|"|+2
Graph f(x) and g(x). Identify all solutions to the equation f(x)=g(x) on the graph.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Item #4
M1.F.IF.C.7 (A1.F.IF.C.8)
Two savings plans are represented by the functions f(x) and g(x) respectively, where x is the number of months and f(x) and g(x) represent the value of each account in dollars.
What is the average rate of change of the higher earning account after 6 months?
x g(x)
1 50
2 200
3 450Write your answer in the space provided.
168
Share one or two “ah-ha” moments from this activity with your neighbor.
Item Assessment ActivityItem #5
A2.A.APR.A.2 (M3.A.APR.A.2)
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.
Identify all zeroes for the following:
$ = 2"' − 5"* − 3", − 2"- + 5" + 3
169
Creating Formative Items
Before you actually start writing items:
• Read the standards carefully with the assessment purpose in mind. Ask yourself: “What skills/knowledge are the standards asking the student to display?”
• Revisit the “I can” statements or “essential questions” you wrote for the standard(s). They may provide guidance as you write items.
• Brainstorm.
Revisiting Standard A1.F.IF.C.7
Determine if each equation will have a minimum or a maximum value. You do not have to provide any coordinates. Match the equations on the left with the correct choice on the top.
FORMATIVE Assessment
A1.F.IF.C.7 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. 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.
Maximum Minimum
$ = (" − 2)-+4
$ = −1(" − 2)-+4
$ = −2"- + 4" − 2
$ = 2"- − 8" + 1
170
• Think about the purpose of the assessment as a whole. Is it formative or summative?
Creating Formative Items
Revisiting Standard A1.F.IF.C.7
Did we cover all aspects of the standard with these items?
Identify the zeroes for the following quadratic equation:
$ = 2"- − 4x + 1
For a quadratic function, complete the square in order to identify the zeroes.
Interpret zeros and extreme values for quadratic equations in terms of a context.
A ball is thrown straight up, from 4 m above the ground with a velocity of 20 m/s. Itsheight is modeled by the following equation:
ℎ = 4 + 208 −58-
How long will it take the ball to hit the ground?
A. 5 secondsB. 4.2 secondsC. 2.2 seconds D. 1 second
Item #2
Item #3
171
Review
• Formative Assessments may need items that scaffold in order for the teacher to diagnose what a student does/does not understand.
• It is very difficult to formatively assess student understanding through a single item.
• Don’t forget the principles of universal design.
• You will be provided a set of standards.
• You and a partner will be writing items to post for our gallery walk.
– On selected response items you do not have to post the rationale for the distractors.
– Please post the coding for the standard(s) to which your items are written.
Item Writing-Your Turn
Selected Response
Multiple Choice Multiple Select
• Effectively writing “I can” or “essential questions” helps target assessment items specifically to standards.
• It’s important to ask yourself the nine essential questions during item review or item writing.
172
Option 1 Option 2
1. Choose three of the standards.2. Write an item to assess each
standard that you would use on a formative assessment.
3. Try to write at least one multiple choice or multiple select item. Focus on writing distractors that provide instructional information.
1. Choose one standard.2. Write three formative assessment
items to the single standard that you selected. Make sure that each item requires students to demonstrate a different level of understanding of the standard.
3. Try to write at least one multiple choice or multiple select item. Focus on writing distractors that provide instructional information.
Item Writing: Your Turn
Use this space to write out your standard(s) and assessment item(s).
173
Gallery WalkAs you review your colleagues’ items, look for similarities and differences in the items created.
• What was challenging about this experience?
• What did you learn from this experience?
• What supports do you need to better understand the relationship between standards and assessments in this way?
Reflection
Reflect on your experience evaluating and creating assessment items and discuss the following:
Notes:
174
Analyzing Assessments
Areas of Focus1. Intent of the Assessment
• Summative
• Formative
2. Content and Structure of Assessments
3. Analysis of Assessments
• Is the data ______________________________________________________________ ?
• How is it analyzed?
• On which questions ________________________________________________________ ? Why?
• On which questions ________________________________________________________ ? Why?
• Were there issues with…
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ ?
Analysis of Assessment
175
Taking Action
• How is instruction changing/adapting as a result of student data?
• Are results shared with all stakeholders (including students)?
• Are assessments adapted to address weaknesses found?
Assessment Instruction
“The assessments will produce no formative benefit if teachers administer them, report the results, and then continue with instruction as previously planned.”
—Stephen and Jan Chappuis, 2012
Notes:
176
Summary
The Cycle of Assessment
Teach
Assess
Analyze
Action
Aligned Materials and Assessments
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
177
Notes:
Appointment with Peers
Please meet with your third partner to discuss the following:
• What are your takeaways from module 7?
• How does this align to your observation rubric?
178
Strong Standards
High Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 4: Assessment and MaterialsModule 8: Evaluating Instructional Materials
179
TAB PAGE
Goals
• Know which key criteria to use for reviewing materials, lessons, and/or units for alignment and quality.
How do we know that our instructional materials address the depth of the content and the instructional shifts of focus, coherence, and rigor of the TN State Standards?
Key Question
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
Instructional Shifts
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Aligned Materials and Assessments
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
180
• Examine the TEAM rubric to define what is meant by standards based materials.
Standards Based Materials and Practice
“…teachers have a responsibility to make day-to-day instructional choices that ensure that students work with problems that engage their interest and their intellect.”
—Cathy L. Seeley, 2014
Reflect on Our Practice
When your students’ work is on public display, in the hallway or shared with families, can anyone see the math?
If anyone looked at your students’ work, would they be able to see the math or would they be left asking “where’s the math?”
Notes:
181
Are the materials and the instructional practices you are using focused on the mathematics?
Standards Based Materials and PracticeTEAM ConnectionActivities & Materials• Support the lesson objective
• Are challenging
• Sustain students’ attention
• Elicit a variety of thinking
• Provide time for reflection
• Provide opportunities for student-to-student interaction
• Provide students with choices
• Incorporate technology
• Induce curiosity & suspense
• In addition sometimes activities are game-like, involve simulations, require creating products, and demand self-direction and self-monitoring.
• The preponderance of activities demand complex thinking and analysis
• Texts & task are appropriately complex
TEAM ConnectionProblem Solving
• Abstraction
• Categorization
• Predicting Outcomes
• Improving Solutions
• Generating Ideas
• Creating & Designing
• Observing & Experimenting
• Drawing Conclusions/Justifying Solutions
• Identify Relevant/Irrelevant Information
182
Standards Based Materials and Practice
Effective Mathematics Teaching Practices1. Establish mathematics goals to focus learning.
2. Implement tasks that promote reasoning and problem solving.
3. Use and connect mathematical representations.
4. Facilitate meaningful mathematical discourse.
5. Pose purposeful questions.
6. Build procedural fluency from conceptual understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking.
• What content standard do you think these activities address?
• Where is the evidence of student understanding of the mathematical content?
Notes:
183
Missing Angle Activity
Standards Based Materials and Practice
Notes:
Missing Angle Activity
If a teacher was trying to address the depth of the content standard 8.G.A.3 , does the Missing Angle Activity accomplish this goal?
8.G.A.3. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
184
Criteria for Alignment and Quality
Research
Notes:
“A curriculum is more than a collection of activities.”
—from the Curriculum Principle in Principles and Standards for School Mathematics
A well-articulated curriculum will:
• Make clear the most important mathematics of the grade level.
• Specify when concepts and skills are introduced and when they should be mastered.
• Detail how student conceptual understanding of big ideas develops across units and across multiple grade levels.
When choosing instructional materials, what should a teacher consider?
185
Criteria for Alignment and Quality
Course Standards Instructional Shifts
Clear Progressions Access for All Students
Quality
Materials Review InstrumentWhen reviewing materials, it is important to have a deep understanding of the standards and a deep understanding of the review instrument before looking at the materials.
• Section I: Non-Negotiable Alignment Criteria
– Part A: Standards
– Part B: Shifts
• Focus
• Rigor
• Coherence
• Section II: Additional Alignment Criteria and Indicators of Quality
– Part A: Key areas of focus
– Part B: Student engagement and instructional focus
– Part C: Monitoring student progress
186
Ma
th M
ate
rials R
ev
iew
Instru
me
nt
SE
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ION
I: NO
N-N
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OT
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ME
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D IN
DIC
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OR
S O
F Q
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LIT
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Pa
rt A.
Co
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Sta
nd
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s
Pa
rt B.
Sh
ifts in In
structio
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Pa
rt A.
Ke
y Are
as o
f Fo
cus
Pa
rt B.
Stu
de
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ag
em
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Instru
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up
po
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Pa
rt C.
Mo
nito
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Stu
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Pro
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ss
Ye
s: Mo
ve to
Pa
rt B
No
: Do
no
t use
or m
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ify
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ve to
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II
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: Do
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: Do
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s: Use
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teria
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No
: Do
no
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or m
od
ify
Th
e in
struc
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al m
ate
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rep
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nt 1
00
pe
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nt
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nm
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the
Te
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esse
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ath
Sta
nd
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an
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ch
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on
the
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stan
da
rds a
t the
rigo
r
ne
ce
ssary
for stu
de
nts to
rea
ch
ma
stery
.
1.
Fo
cus
2.
Co
he
ren
ce
3.
Rig
or
Le
arn
ing
ex
pe
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rovid
e
op
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rtun
ities fo
r tho
ug
ht,
disco
urse
, an
d p
ractice
in a
n
inte
rcon
ne
cted
an
d so
cial
con
tex
t.
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its an
d in
structio
na
l
seq
ue
nce
s are
coh
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nt a
nd
org
an
ized
in a
log
ical m
an
ne
r
tha
t bu
ilds u
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n k
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wle
dg
e
an
d sk
ills lea
rne
d in
prio
r gra
de
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ls or e
arlie
r in th
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Ma
teria
ls sup
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t
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with
in a
n E
LA
focu
s by p
rovid
ing
con
sisten
t
op
po
rtun
ities fo
r stud
en
ts to
utilize
litera
cy sk
ills for
pro
ficien
cy in re
ad
ing
, writin
g,
voca
bu
lary, sp
ea
kin
g, a
nd
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.
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teria
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vide
s lea
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at in
corp
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ard
s,
Sta
nd
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s for M
ath
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atica
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Pra
ctice, a
nd
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racy S
kills for
Ma
the
ma
tical P
roficie
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teria
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s stud
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, rele
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L, h
ave
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rform
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low
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de
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teria
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de
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d
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teria
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vide
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for stu
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ll as e
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nsio
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for stu
de
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dy m
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ting
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stery o
r with
hig
h in
tere
st.
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ssme
nts p
rovid
e d
ata
on
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ten
t stan
da
rds.
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teria
l asse
sses stu
de
nt
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sing
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ds th
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de
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teria
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de
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terp
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an
ce.
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teria
l use
s varie
d m
od
es o
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lum
em
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dd
ed
asse
ssme
nts th
at m
ay in
clud
e
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ative
, sum
ma
tive,
an
d se
lf-asse
ssme
nt m
ea
sure
s.
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ssme
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re e
mb
ed
de
d
thro
ug
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ut in
structio
na
l
ma
teria
ls as to
ols fo
r stud
en
ts’
lea
rnin
g a
nd
tea
che
rs’
mo
nito
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of in
structio
n.
Asse
ssme
nts p
rovid
e te
ach
ers
with
a ra
ng
e o
f da
ta to
info
rm
instru
ction
.
Evaluating Instructional Materials: Best Practices
• It’s important to review instructional materials you use to determine where you have strong alignment to standards and where you may have gaps to fill.
• School leaders and teachers should engage in reviewing instructional materials on an ongoing basis to develop pedagogy and capacity.
• There is a new adoption.
• Current materials have gaps that may require supplemental materials.
• They are looking for supplemental instructional materials.
Notes:
188
Teachers need to review materials when:
Supplemental Materials
Let’s Discuss:
• What resources do you have on hand?
• Where do you find supplemental materials?
• How can you use this process to evaluate supplemental materials?
Notes:
Reviewing Materials: A RecapAs you look for materials…
• Is it aligned to the standards?
• Does it reflect high leverage best practices?
• Is it accessible for ALL students?
• Does it lead to students being able to demonstrate mastery of the standard?
189
Potential Gaps in Materials
Grades 6-8:
• Shifted Compound Probability standard
• Moved from seventh to eighth grade
• Revised Geometry standards
• Removed from seventh grade: slice of 3-dimensional objects
• Removed from eighth grade: congruency and similarity of 2-dimensional objects
Grades 9-12:
• Shifted a number of standards from Algebra II and Integrated Math III to the Additional Math Courses
Notes:
190
Module 8 Review
Aligned Materials and Assessments
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
The review process of instructional materials will:
alignment or gaps, and
§ Result in wise decisions about how best to use the materials already on-site to teach the new standards to mastery OR effectively fill any gaps uncovered in the review process.
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§ Deepen understanding of the standards,
§ Make use of review instruments to analyze materials to determine
Notes:
Appointment with Peers
Please meet with your fourth partner to discuss the following:
• How can the materials review instrument lead to improved student outcomes in your classroom?
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Strong Standards
High Expectations
Instructional ShiftsAligned
Materials and Assessments
Part 5: Putting It All TogetherModule 9: Instructional Planning
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TAB PAGE
Goals
• Develop lesson planning techniques to strengthen the understanding of the relationship between standards and practice.
• Create lessons based on the revised standards to be used for instruction.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Instructional Shifts
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
Aligned Materials and Assessments
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
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• Understand intentional instruction as a bridge between strong standards and assessment.
Designing Effective Learning Experiences
“…teachers have a responsibility to make day-to-day instructional choices that ensure that students work with problems that engage their interest and their intellect.”
—Cathy L. Seeley, 2014
Notes:
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Designing Effective Learning Experiences
What is intentional instruction?
Notes:
Know your standard
Essential Questions or I
Can’s
KUD
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• Standards are driving your instruction.• What does "intentional" mean?• Gather evidence of learning (assessments)• What are your end goals?
Designing Effective Learning ExperiencesPutting It All Together
Now it’s your turn! Use this space for your small group planning.
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• We brought together concepts studied in the previous modules to plan for instruction.
• We were intentional in our instruction, using instruction to bridge the gap between standards and assessment.
Intentionally Planned
Instruction
Launch, Explore, Summarize
Understanding by Design
Your Method
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There are many ways to intentionally plan instruction.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
Shifts in Instructional Practice
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Aligned Materials and Assessment
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
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Module 9 Review
• There are many ways to “do” intentional instruction.• Intentional instruction is the bridge between standards and assessment.• Start with the standard: determine what students need to know, understand, and
do.• Create learning experiences that connect to students’ experiences and give them a
chance to explore the concept.
High Expectations
We have a continued goal to prepare students to be college and career ready.
Strong Standards
Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.
Shifts in Instructional Practice
The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.
Aligned Materials and Assessment
Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.
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• Assessment plays a critical role in instruction, should be standards based, and should be used to determine student mastery of the standard(s).
“Districts and schools in Tennessee will exemplify excellence and equity such that
all students are equipped with the knowledge and skills to successfully
embark upon their chosen path in life.”
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