Common Core State Standards
and Assessments
Grades 9 - 12
Mathematics
Training of the Trainers
July 2012
Work Session Answer Keys
1
PARCC Model Content
Frameworks for Mathematics
(pp 2 -3)
Individual End of Course Overviews:
• Shows which standards are assessed on a given end-of-course assessment as well as relative
cluster emphases.
• Neglecting material will leave gaps in student skill and understanding and may leave
students unprepared for the challenges of a later course.
• Algebra I (pp. 6 - 7)
• Geometry (pp 11 – 12)
•
Algebra II (pp 15 – 17)
•
Major Content
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Supporting Content
Additional Content
• Underlined numerals indicate standards eligible for assessment on two or more end-of
course assessments—cross cutting standards.
2
PARCC Model Content
Frameworks for Mathematics
(page 3)
Examples of Key Advances from Previous Grades or Courses:
• Highlights some of the major steps in the progression of increasing
knowledge and skill from year to year.
• Each key advance in the mathematical content also corresponds to a
widening scope of problems that students can solve.
Discussion of Mathematical Practices in Relation to Course
Content:
• Highlights some of the mathematical practices and describes how they
play a role in each course.
• These examples are provided to stress the need to connect content and
practices, as required by the standards.
3
PARCC Model Content
Frameworks for Mathematics
(page 4)
Fluency Recommendations:
• The high school standards do not set explicit expectations for
fluency nor will the PARCC assessments address fluency, but fluency
is important in high school mathematics.
• Fluency can allow for smooth progress beyond the CCR threshold
toward readiness for further study/careers in STEM fields.
• Fluency is not meant to come at the expense of understanding; it is
an outcome of a progression of learning and thoughtful practice.
• Note: Fluency as defined by PARCC means to flow without
halting, stumbling or reversing; and, it marks the endpoints of
progressions of learning. (PARCC Model Content Frameworks (Grades K – 8, page 8)
4
PARCC Model Content
Frameworks for Mathematics
(pp 4 - 5)
Pathway Summary Tables:
(Table 1 and Table 3)
•
Table 1: Traditional Pathway (page 20)
•
Table 3: Integrated Pathway (page 35)
•
Shows three end-of-course assessments’ standards at a glance.
•
Standards shown by a dot () will be assessed in that course.
•
Shading indicates standards that are appropriate for more than one
end-of-course assessment.
•
Is consistent with the preceding end-of-course overviews.
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5
PARCC Model Content
Frameworks for Mathematics
(pp 4 - 5)
Assessment Limits Tables:
(Table 2 and Table 4)
• Table 2: Traditional Pathway (pp 21 – 24)
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• Table 4: Integrated Pathway (pp 36 – 42)
• Specifies how a standard that is appropriate for more than one
end-of-course test will be assessed.
• The number of standards assessed on more than one end-ofcourse assessment has been kept as small as possible, so as to
keep the number of standards in each course from becoming
too large.
6
PARCC Model Content
Frameworks for Mathematics
(pp 43 - 45)
Examples of Opportunities for Connections among Standards,
Clusters, Domains, or Conceptual Categories:
The standards identify a number of connections among conceptual
categories; and, connections among standards, clusters, and domains.
Examples of Opportunities for Connecting Mathematical
Content and Mathematical Practices:
Identifies opportunities for connecting the CCSS to individual Standards
for Mathematical Practice.
7
Referencing the CCSS for
Mathematics
Implications for Instruction for F-IF.4
Sample Responses:
• Cross-cutting standard: The Algebra I teacher and Algebra II teacher must
proactively plan together to scaffold learning.
• Assessment limits: Assessment tasks will have a real-world context.
Assessment tasks in Algebra II may include logarithmic and trigonometric
functions.
• Major Content Emphasis in both Algebra I and Algebra II: This
standard (along with other standards) will constitute a majority of the
assessment; both teachers must ensure fluency of skills and conceptual
understanding. May be taught in conjunction with parts of F-IF.7.
• Modeling standard: Both teachers must provide real-world examples and
contexts in their instruction.
8
Implications for Instruction for N-VM.5
Sample Response:
Plus standard: This standard will not appear in
the common math curriculum for all students
(Algebra I, Algebra II, and Geometry). It will be
taught in our fourth year course Common Core
Plus. Teachers can be creative in its presentation
and assessment until the final version of the
PARCC MCF is released.
*Note: refer to PARCC Model Content Frameworks for High School Mathematics page 5.
9
Work Session 2
CCSS for Mathematics
“Scavenger Hunt”
#1. These standards do not dictate curriculum or teaching methods.
For example, just because topic A appears before topic B in the
standards for a given grade, it does not necessarily mean that topic
A must be taught before topic B. A teacher might prefer to teach
topic B before topic A, or might choose to highlight connections
by teaching topic A and topic B at the same time.
#3. Where can a teacher find a list of terms and their definitions for
commonly used words in the CCSS for Mathematics? The
Glossary On what page(s) can this information be found? Pages
85-90
#4. T or F. There are only two plus standards under the Algebra
Conceptual Category.
10
Work Session 2
continued
#5.
According to the Introduction for Geometry,
connections should be made to equations.
#6. What are the two “cluster headings” for S-MD?
(1) Calculate expected values and use them to
solve problems.
(2) Use probability to evaluate outcomes of
decisions.
#7. Using your response from question #6, what is
important to note about the standards listed under this
particular domain (MD)?
All of these standards are plus standards.
11
Work Session 2
continued
#8.
On what page can N-Q.3 be found? Page 60
#9.
Reference the standard listed below: G-GMD.2
#10. Find one person in your group that knows Cavalieri’s principle
and summarize it below:
Given two solids of equal altitude, if their cross sections
are made by parallel planes at the same distance from
their respective bases, then the volumes of the two
solids are equal.
#11. Locate and read the standard reference by F-BF.2. What is
important to note about this standard?
This is a Modeling standard.
12
Work Session 2
continued
#12. Based on your response to question #11,
would all students be taught this standard?
Yes
Justify your response.
All standards without a (+) symbol
appear in the common math
curriculum (Algebra I, Algebra II, and
Geometry) for all students.
13
Work Session 2
continued
#13.
Compare and contrast A-CED.1 and MMF Algebra I
Objective 2a using the Venn Diagram below.
create
rational,
quadratic and
exponential
functions
CCSS A-CED.1
linear
equations and
inequalities
solve
real world
application
assessed in
Algebra I
graph
rational
coefficients
check
Algebra I Obj. 2a
14
Work Session 2
continued
#14.
Given the data set below, do what is expected of a
mathematically proficient student as it relates to standards
S-ID. 7 and S-ID.8.
Cybal’s Summer Exercise
Program
Week
Pounds Lost
5
17
1
8
3
13
4
15
•Slope/rate of change = 2.26 = Cybal lost
approximately 2.26 pounds each week.
•Y-intercept = 5.9 =Cybal lost approximately
6 pounds at the beginning of her summer
exercise program.
•Correlation Coefficient = r = 0.998
Interpretation: There is an almost perfect,
positive linear relationship between the week
and number of pounds that Cybal lost.
Also, as the number of weeks increased so did
the number of pounds Cybal lost.
15
Work Session 5a:
Focusing on a High School CCSS
(A-REI.11)
#3. Write one Essential Question for this
standard.
Sample Responses:
(1) How can you determine the
most efficient way for solving a
system of equations?
(2) How can the value of the
constants of a function be used to
determine if a solution exists for a
system of equations?
16
Work Session 5a
continued
#4. How can the Modeling diagram be applied to
this standard?
Sample Response:
Bowl-o-Rama charges $2.50 per
game plus $2 for shoe rental, and
Bowling Pinz charges $2 per game
plus $4 for shoe rental. For how
many games will the cost to bowl be
the same at both places?
17
Work Session 5a
continued
This real world problem can
be modeled by the system
of equations:
c = 2.5g + 2
c = 2g + 4
18
Work Session 5a
continued
Create a graph of
the system on
the same
coordinate plane
Solve the system
of equations
algebraically
Compare the xvalue of the
solution to the
graph drawn AND
substitute into the
original equations
Report the solution
in the context of
the problem and as
the only value(s)
that make both
equations true.
Explain what the
solution
represents
19
Work Session 5a
continued
#5. Identify the cluster heading for
this standard.
Represent and solve equations
and inequalities graphically
20
Work Session 6a:
Focusing on a High School CCSS
(G-SRT.2)
#3. List synonyms for the key term “transformation.”
Sample Response:
relocate, move, shift, slide, glide, and switch.
#4. Write one “I Can” statement for this standard.
Sample Response:
“I can, by using similarity transformations,
explain that triangles are similar if all pairs of
corresponding angles are congruent and all
corresponding pairs of sides are proportional.”
21
Work Session 6a
continued
#5. Indicate one instructional method a
teacher may use to ensure that
Mathematical Practice #3 is incorporated
in this Standard.
Sample Responses:
The teacher could set up “Learning
Centers”, “Work Stations”, or
“Think-Pair-Share” activities.
22
Work Session 7a:
Focusing on a High School CCSS
(F-BF.1a)
#3. Identify the prerequisite skills that are mentioned in 8.F.1 for this
standard.
Sample Responses:
(1) understand what a function is
(2) utilize a function rule
(3) graph a function
(4) graph a set of ordered pairs
#4.
Locate and reference the MMF Pre-Calculus objective that closely
matches this standard.
Objective 1a: Express sequences and series using recursive and
explicit formulas. (DOK 2)
#5.
Locate and reference the MMF Discrete Mathematics objective that
closely matches this standard.
Objective 2e: Define a sequence recursively and explicitly. (DOK 2)23
Work Session 7b:
Focusing on a High School CCSS
(F-BF.1a)
#2. Complete the chart below.
(Note: Open Excel file 9-12 WS #7b Chart)
#9. Determine the function rule for Mr.
Donald’s commission.
c = 100  2(n-1)
#10. Determine the function rule for
Larry’s salary at the end of each day.
s = 2(n – 1)  (1000 – 100n)
24
Work Session 7b
continued
Number of Days
Larry’s Salary at the End
of the Day
1
20 · 1000 – 20 · 100
2
2 · (20 · 1000 – 20 · 100) – 21 · 100
21 · 1000 – 21 · 100 – 21 · 100
21 · 1000 – 2 · 21 · 100
3
2 · (21 · 1000 – 2 · 21 · 100) – 22 · 100
22 · 1000 – 2 · 22 · 100 – 22 · 100
22 · 1000 – 3 · 22 · 100
:
:
n
2n – 1 · 1000 – n · 2n – 1 · 100
2n – 1 · (1000 – 100n)
25