### The CCSS for Mathematical Practice

```Supporting Rigorous Mathematics
Teaching and Learning
Deepening Our Understanding of CCSS Via
A Constructed Response Assessment
Tennessee Department of Education
High School Mathematics
Algebra 2
Forms of Assessment
Assessment as Learning
Assessment of Learning
Assessment for Learning
Session Goals
Participants will:
• deepen understanding of the Common Core State
Standards (CCSS) for Mathematical Practice and
Mathematical Content;
• understand how Constructed Response
Assessments (CRAs) assess the CCSS for both
Mathematical Content and Practice; and
• understand the ways in which CRAs assess
students’ conceptual understanding.
Overview of Activities
Participants will:
• analyze Constructed Response Assessments
(CRAs) in order to determine the way they are
assessing the CCSSM;
• analyze and discuss the CCSS for Mathematical
Content and Mathematical Practice;
• discuss what it means to develop and assess
conceptual understanding; and
• discuss the CCSS related to the tasks and the
implications for instruction and learning.
The Common Core State Standards
The standards consist of:
 The CCSS for Mathematical Content
 The CCSS for Mathematical Practice
Analyzing a
Performance-Based Assessment
Tennessee Focus Clusters
Algebra 2
 Extend the properties of exponents to rational
exponents.
 Write expressions in equivalent forms to solve
problems.
 Understand the relationship between zeros and
factors of polynomials.
 Build a function that models a relationship between
two quantities.
7
Analyzing Assessment Items
(Private Think Time)
Four assessment items have been provided:
 Car Depreciation
 Writing a Polynomial
 Patterns in Patterns
 One Rocket, Three Equations
For each assessment item:
• solve the assessment item; and
• make connections between the standard(s) and the
assessment item.
1. Car Depreciation
After you purchase a new car, it begins to lose value. As the
years pass, more and more value is lost. This process is
known as depreciation.
a. Carmen buys a new car for \$24,500. Carmen’s new car
loses 14% of its value each year through depreciation.
Write a function that can be used to model the value of the
car at the end of each year that Carmen owns the car.
b. After Carmen buys the car, he adds an audio and speaker
system worth \$500. The audio system loses 15% of its
value the minute it is installed into the car, and 7% of the
remaining value each year through depreciation. Write a
function that can be used to model the value of the car
with the audio system and speakers at the end of each
year that Carmen owns the car.
2. Writing a Polynomial
Recall that polynomial functions with only real number zeros can be
written in factored form as follows:
y  a   x  z1  1   x  z 2  2  ...   x  z n 
p
p
pn
where each zn represents some real root of the function, and each pn
is a whole number exponent greater than or equal to 1.
Consider the graph of the polynomial function below.
14
a. Lisa claims that, since the point (0, 6)
13
12
is on the graph, (x – 6) is a factor of
11
10
this polynomial. Explain why you
9
8
agree or disagree with Lisa’s claim.
7
Identify all the zeroes of the function
6
5
and use that information in your
4
3
explanation.
2
1
1
b. Suppose a = . Write a function in
0
12
-5
-4
-3
-2
-1 -1 0
1
2
3
4
factored form to represent this graph.
-2
5
3. Patterns in Patterns
Laura creates a design of circles embedded in each
other for a poster. The largest circle has a diameter of
28 inches, and each successive circle has a diameter of
3
the previous circle.
4
a. Write a function that can be used to
determine the diameter of any circle
drawn in the poster in this way. Explain
the meaning of each term in your
expression in the context of the
problem.
b. Laura eventually draws 10 circles. Write
and use a formula for the sum of a
series to find the sum of the
circumferences of the 10 circles,
accurate to two decimal places. Show
28 inches
4. One Rocket, Three Equations
For his science project, Grady designs and launches a model
rocket from the rooftop of an empty building. He places a
sensor on the rocket to provide him with data about the height
of the rocket (from ground level in feet) over time (in seconds)
during the flight.
From the data returned by the sensor, Grady was able to write
and graph a function modeling the height of the rocket over
time.
= −16 2 + 96 + 112
a. Rewrite the function in vertex form. Then describe what this
form of the function reveals in the context of the situation.
b. Rewrite the function by factoring completely. Then describe
what this form of the function reveals in the context of the
situation.
Discussing Content Standards
(Small Group Time)
For each assessment item:
With your small group, discuss the connections
between the content standard(s) and the assessment
item.
Deepening Understanding of the Content
Standards via the Assessment Items
(Whole Group)
As a result of looking at the assessment items, what
do you better understand about the specifics of the
content standards?
What are you still wondering about?
The CCSS for Mathematical Content
CCSS Conceptual Category – Number and Quantity
The Real Number System
(N-RN)
Extend the properties of exponents to rational exponents.
N-RN.A.1 Explain how the definition of the meaning of rational
exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for
radicals in terms of rational exponents. For example, we
define 51/3 to be the cube root of 5 because we want (51/3)3
= 5(1/3)3 to hold, so (51/3)3 must equal 5.
N-RN.A.2 Rewrite expressions involving radicals and rational
exponents using the properties of exponents.
Common Core State Standards, 2010
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Seeing Structure in Expressions
(A-SSE)
Write expressions in equivalent forms to solve problems
A-SSE.B.3
Choose and produce an equivalent form of an
expression to reveal and explain properties of the
quantity represented by an expression. ★
A-SSE.B.3c Use the properties of exponents to transform
expressions for exponential functions. For example the
expression 1.15t can be rewritten as (1.151/12)12t ≈
1.01212t to reveal the approximate equivalent monthly
interest rate if the annual rate is 15%.
A-SSE.B.4 Derive the formula for the sum of a finite geometric
series (when the common ratio is not 1), and use the
formula to solve problems. For example, calculate
mortgage payments.★
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire
domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Arithmetic with Polynomials and Rational Expressions
(A-APR)
Understand the relationship between zeros and factors of
polynomials
A-APR.B.2 Know and apply the Remainder Theorem: For a
polynomial p(x) and a number a, the remainder on
division by x – a is p(a), so p(a) = 0 if and only if (x – a)
is a factor of p(x).
A-APR.B.3 Identify zeros of polynomials when suitable
factorizations are available, and use the zeros to
construct a rough graph of the function defined by the
polynomial.
Common Core State Standards, 2010
The CCSS for Mathematical Content
CCSS Conceptual Category – Functions
Building Functions
(F-BF)
Build a function that models a relationship between two
quantities
F-BF.A.1
Write a function that describes a relationship between two
quantities.★
F-BF.A.1a Determine an explicit expression, a recursive process, or
steps for calculation from a context.
F-BF.A.1b Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of
a cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model.
F-BF.A.2 Write arithmetic and geometric sequences both recursively
and with an explicit formula, use them to model situations,
and translate between the two forms.★
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire
domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010
Determining the Standards for
Mathematical Practice Associated with
the Constructed Response Assessment
Getting Familiar with the CCSS for
Mathematical Practice
(Private Think Time)
• Count off by 8. Each person reads one of the CCSS
for Mathematical Practice.
prepared to share the “gist” of the Mathematical
Practice.
20
The CCSS for Mathematical Practice
1.
Make sense of problems and persevere in solving
them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning
of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO
21
Discussing Practice Standards
(Small Group Time)
Each person has 2 minutes to share important
Practice.
22
Discussing Practice Standards
(Small Group Time)
For each assessment item:
With your small group, discuss the connections
between the practice standards and the assessment
item.
The CCSS for Mathematical Practice
1.
Make sense of problems and persevere in solving
them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the
reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated
reasoning.
Common Core State Standards, 2010
Deepening Understanding of the Practice
Standards via the Assessment Items
(Whole Group)
Which Standards for Mathematical Practice do you
better understand?
What are you still wondering about?
Assessing Conceptual Understanding
Rationale
We have now examined assessment items and
discussed their connection to the CCSS for
Mathematical Content and Practice. A question that
needs considering, however, is if and how these
assessments will give us a good means of measuring
the conceptual understandings our students have
acquired.
In this activity, you will have an opportunity to
consider what it means to develop conceptual
understanding, as described in the CCSS for
Mathematics, and what it takes to assess for it.
Assessing for Conceptual Understanding
The set of CRA items are designed to assess student
understanding of expressions and equations.
Look across the set of related items. What might a
teacher learn about a student’s understanding by
looking at the student’s performance across the set of
items as a whole?
What is varying from one item to the next?
Conceptual Understanding
• What do the authors mean by conceptual
understanding?
• How might analyzing student performance on this
set of assessments help us determine if students
have a deep understanding of the assessed
standards?
Developing Conceptual Understanding
Knowledge that has been learned with understanding
provides the basis of generating new knowledge and
for solving new and unfamiliar problems. When
students have acquired conceptual understanding in
an area of mathematics, they see connections among
concepts and procedures and can give arguments to
explain why some facts are consequences of others.
They gain confidence, which then provides a base
from which they can move to another level of
understanding.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics.
The CCSS on Conceptual Understanding
In this respect, those content standards which set an
expectation of understanding are potential “points of
intersection” between the Standards for Mathematical
Content and the Standards for Mathematical Practice.
These points of intersection are intended to be weighted
toward central and generative concepts in the school
mathematics curriculum that most merit the time,
resources, innovative energies, and focus necessary to
qualitatively improve the curriculum, instruction,
assessment, professional development, and student
achievement in mathematics.
Common Core State Standards for Mathematics, 2010, p. 8, NGA Center/CCSSO
Assessing Concept Image
Tall (1992) differentiates between the mathematical definition of a
concept and the concept image, which is the entire cognitive
structure that a person has formed related to the concept. This
concept image is made up of pictures, examples and non-examples,
processes, and properties.
A strong concept image is a rich, integrated, mental representation
that allows the student to flexibly move between multiple
formulations and representations of an idea. A student who has
connected mathematical ideas in this way can create and use a
model to analyze a situation, uncover patterns and synthesize them
to form an integrated picture. They can also use symbols
meaningfully to describe generalizations which then provides a base
from which they can move to another level of understanding.
Brown, Seidelmann, & Zimmermann. In the trenches: Three teachers’ perspectives on moving beyond the math wars.
http://mathematicallysane.com/analysis/trenches.asp
Developing and Assessing Understanding
Why is it important, when assessing a student’s
conceptual understanding, to vary items in these
ways?
Using the Assessment to Think About
Instruction
In order for students to perform well on the CRA, what
are the implications for instruction?
• What kinds of instructional tasks will need to be
used in the classroom?
• What will teaching and learning look like and sound
like in the classroom?
Step Back
• What have you learned about the CCSS for
Mathematical Content that surprised you?
• What is the difference between the CCSS for
Mathematical Content and the CCSS for
Mathematical Practice?
• Why do we say that students must work on both
Standards for Mathematical Content and the
Standards for Mathematical Practice?
Seeing Structure in Expressions
A container of ice cream is taken from the freezer and sits in a room for
t minutes. Its temperature in degrees Fahrenheit is  −  2
−
+ ,
where  and  are positive constants. Write this expression in a form
that shows that the temperature is always
1. Less than  +
2. Greater than
A.SSE.3
http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf, pg. 5
Seeing Structure in Expressions
Susan is told to take 250 mg doses of an antibiotic for an ear infection
regularly every 12 hours for 10 days. Susan does some research
online and finds out that at the end of 12 hours, about 4% of the drug is
still in the body.
1. What quantity of the drug is in the body right after the first dose, the
second dose, the third dose?
2. When will the total amount of the antibiotic in Susan’s body be the
highest? What is that amount?
A.SSE.B.4
http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf, pg. 5
Seeing Structure in Expressions
1. What quantity of the drug is in the body right after the first dose, the
second dose, the third dose?
Let 1 be the amount of the drug in the body after the first dose
1 = 250
To find 2 , the amount of the drug in the body after the second dose,
we have to find out how much of the first dose is still present and add
the new dose. Since after 12 hours about 4% of the drug is still present
in the body we get:
2 = 0.041 + 250 = 0.04 250 + 250 = 260
We use the same reasoning to find 3 :
3 = 0.042 + 250 = 0.04 0.04 250 + 250 + 250 = 260.4
Or
3 = 0.042 1 + 0.041 + 1
http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf, pg. 5
Seeing Structure in Expressions
2. When will the total amount of the antibiotic in Susan’s body be the
highest? What is that amount?
Since the amount of antibiotic in Susan’s body increases every time she
takes a dose, we need to find the amount in her body after the last (20th)
dose:
20 = 0.0419 1 + 0.0418 1 + ⋯ + 0.041 + 1
20 = 1 0.0419 + 0.0418 + ⋯ + 0.04 + 1
20 =
1 1−0.0420
1−0.04
or 20 =
1 −1 0.04 20
1−0.04
Therefore, the amount of antibiotic in Susan’s system after 20 does of
250 mg is:
250−250 0.04 20
20 =
=260.4167 mg ≈ 260.4 mg pg. 5
http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf,
1−0.04
1. Car Depreciation
Building Functions
(F-BF)
Build a function that models a relationship between two
quantities
F-BF.A.1
Write a function that describes a relationship between two
quantities.★
F-BF.A.1b Combine standard function types using arithmetic
operations. For example, build a function that models the
temperature of a cooling body by adding a constant
function to a decaying exponential, and relate these
functions to the model.
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain
is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010
2. Writing a Polynomial
Arithmetic with Polynomials and Rational Expressions
(A-APR)
Understand the relationship between zeros and factors of
polynomials
A-APR.B.3 Identify zeros of polynomials when suitable factorizations
are available, and use the zeros to construct a rough graph
of the function defined by the polynomial.
Building Functions
(F-BF)
Build a function that models a relationship between two
quantities
F-BF.A.1
Write a function that describes a relationship between two
quantities.★
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire
domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010
Building Functions
(F-BF)
Build a function that models a relationship between two
quantities
F-BF.A.2
Write arithmetic and geometric sequences both
recursively and with an explicit formula, use them to
model situations, and translate between the two
forms.
Seeing Structure in Expressions
A-SSE
Write expressions in equivalent forms to solve problems
A.SSE.B.4 Derive the formula for the sum of a finite geometric
series (when the common ratio is not 1), and use the
formula to solve problems. For example, calculate
mortgage payments.★
★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and
specific modeling standards appear throughout the high school standards indicated with a star ( ★). Where an
entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010
4. One Rocket, Three Equations
Seeing Structure in Expressions
A-SSE
Write expressions in equivalent forms to solve problems
A.SSE.B.3 Choose and produce an equivalent form of an
expression to reveal and explain properties of the
quantity represented by an expression. ★
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and
specific modeling standards appear throughout the high school standards indicated with a star (★). Where an
entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010
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