Supporting Rigorous Mathematics
Teaching and Learning
Deepening our Understanding of the CCSS Via
a Constructed Response Assessment
Tennessee Department of Education
High School Mathematics
Algebra 1
© 2013 UNIVERSITY OF PITTSBURGH
Forms of Assessment
Assessment as Learning
Assessment of Learning
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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.
© 2013 UNIVERSITY OF PITTSBURGH
Overview of Activities
Participants will:
• analyze Constructed Response (CRAs) in order to
determine the way they are assessing the CCSS for
Mathematics;
• analyze and discuss the CCSS for Mathematical
Content and CCSS for 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.
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The Common Core State Standards
The standards consist of:
 The CCSS for Mathematical Content
 The CCSS for Mathematical Practice
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Analyzing a
Constructed Response Assessment
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Tennessee Focus Clusters
Algebra 1
 Create equations that describe numbers or
relationships.
 Solve equations and inequalities in one variable.
 Represent and solve equations and inequalities
graphically.
 Interpret functions that arise in applications in terms
of the context.
© 2013 UNIVERSITY OF PITTSBURGH
Analyzing Assessment Items
(Private Think Time)
Four assessment items have been provided:
 Buddy Bags
 Disc Jockey Decisions
 What’s the point?
 Paulie’s Pen
For each assessment item:
• solve the assessment item; and
• make connections between the standard(s) and the
assessment item.
© 2013 UNIVERSITY OF PITTSBURGH
1. Buddy Bags
For a student council fundraiser, Anna and Bobby have spent a total
of $55.00 on supplies to create Buddy Bags. They plan to charge
$2.00 per Buddy Bag sold.
Anna created the graph below from an equation to represent the
profit from the number of Buddy Bags sold.
a. Determine the equation Anna used to create the
graph if x represents the number of Buddy Bags
sold and y represents the profit in dollars. Use
mathematical reasoning to explain your
equation.
c. Anna says, “I connected the points to represent
the equation, but by connecting the points I am
not representing the context of the problem.”
Use mathematical reasoning to explain why she
is correct.
© 2013 UNIVERSITY OF PITTSBURGH
60
50
40
30
Profit in Dollars
b. Bobby claims that Anna’s graph is incorrect
because it does not show that they plan to
charge $2.00 per Buddy Bag. Do you agree or
disagree with Bobby? Use mathematical
reasoning to support your decision.
70
20
10
0
-10
0
10
20
30
40
50
-20
-30
-40
-50
-60
Number of Buddy Bags Sold
60
2. Disc Jockey Decisions
The student council has asked Dion to be the disc jockey for the Fall
Banquet. He has been asked to play instrumental music during the first
hour while the students are eating dinner. During the last 15 minutes of the
banquet the school choir will sing. For the remaining time, Dion will choose
popular songs to play.
a. Write an equation to determine the number of popular songs, p, that
Dion can choose if the songs Dion chooses have an average run time
of 3.5 minutes and the total time for the Fall Banquet is t minutes. Use
mathematical reasoning to justify that your equation is correct.
b. Use your equation from Part a to determine the number of popular
songs that Dion can choose if the banquet will be held from 6:00 –
10:00pm.
c. Dion decides to organize the music another way. He decides to play 50
popular songs. Write and solve an algebraic equation to determine the
average run time, r, of the 50 popular songs Dion can choose if the
average run time is represented in minutes by r. Use mathematical
reasoning to justify that your equation is correct.
© 2013 UNIVERSITY OF PITTSBURGH
3. What’s the point?
Mr. Williams asks his Algebra 1 class to find the solutions to an
equation in two variables with a domain in the set of real numbers.
Colton correctly creates the table below using values from the
domain of the equation. He then uses his table to create a graph.
Colton’s Table
x
y
0
-4
2
-1.5
-2
-6.5
5
2.25
-8
-14
10
8.5
Colton's Graph
10
5
0
-10
-5
0
5
10
-5
-10
-15
a. Determine the equation Colton used to create the table. Use
mathematical reasoning to justify that the equation is correct.
b. Destiny sees Colton’s work and argues that any table contains
just a subset of the solutions to Mr. Williams’ equation. Do you
agree or disagree with Destiny? Explain why or why not.
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4. Paulie’s Pen
Scarlet has to build a rectangular pen for her peacock,
Paulie. Scarlet has P feet of fencing.
a. Scarlet decides the length of the pen will be 12 feet.
Write an equation describing the relationship between P
and the width of the pen, w. Explain your thinking in the
context of the problem.
b. Solve the equation for w. Show your work.
c. Scarlet changes her mind and decides that the width of
Paulie’s pen will be half of the length of his pen. Write
an equation describing the relationship between P and
the width of the pen, w. Explain your thinking in the
context of the problem.
d. Solve the equation for w. Show your work.
© 2013 UNIVERSITY OF PITTSBURGH
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.
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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?
© 2013 UNIVERSITY OF PITTSBURGH
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Creating Equations★
(A –CED)
Create equations that describe numbers or relationships
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 simple rational and exponential functions.
A-CED.A.2 Create
equations in two or more variables to represent
relationships between quantities; graph equations on coordinate
axes with labels and scales.
A-CED.A.3 Represent constraints by equations or inequalities, and by systems
of equations and/or inequalities, and interpret solutions as viable or
nonviable options in a modeling context. For example, represent
inequalities describing nutritional and cost constraints on
combinations of different foods.
A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations. For example, rearrange
Ohm’s law V = IR to highlight resistance R.
★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
Reasoning with Equations and Inequalities
(A-REI)
Solve equations and inequalities in one variable
A.-REI.B.3
Solve linear equations and inequalities in one variable,
including equations with coefficients represented by letters.
A.-REI.B.4
Solve quadratic equations in one variable.
A.-REI.B.4a Use the method of completing the square to transform any
quadratic equation in x into an equation of the form (x – p)2 = q
that has the same solutions. Derive the quadratic formula from
this form.
A.-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49),
taking square roots, completing the square, the quadratic
formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a
and b.
Common Core State Standards, 2010
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Reasoning with Equations and Inequalities
(A-REI)
Represent and solve equations and inequalities graphically
A.-REI.D.10
Understand that the graph of an equation in two variables is the set of
all its solutions plotted in the coordinate plane, often forming a curve
(which could be a line).
A.-REI.D.11
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 solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x) are
linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.★
A.-REI.D.12
Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and
graph the solution set to a system of linear inequalities in two
variables as the intersection of the corresponding half-planes.
★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 – Functions
Interpreting Functions
(F-IF)
Interpret functions that arise in applications in terms of the context
F-IF.B.4
For a function that models a relationship between two quantities, interpret
key features of graphs and tables in terms of the quantities, and sketch
graphs showing key features given a verbal description of the relationship.
Key features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.★.
F-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n)
gives the number of person-hours it takes to assemble n engines in a
factory, then the positive integers would be an appropriate domain for the
function.★
F-IF.B.6
Calculate and interpret the average rate of change of a function
(presented symbolically or as a table) over a specified interval. Estimate
the rate of change from a graph.★
★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
© 2013 UNIVERSITY OF PITTSBURGH
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.
• Read your assigned Mathematical Practice. Be
prepared to share the “gist” of the Mathematical
Practice.
© 2013 UNIVERSITY OF PITTSBURGH
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
information about his/her assigned Mathematical
Practice.
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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.
© 2013 UNIVERSITY OF PITTSBURGH
The CCSS for Mathematical Practice
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving
them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the
reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
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?
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Assessing Conceptual Understanding
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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.
© 2013 UNIVERSITY OF PITTSBURGH
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?
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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?
© 2013 UNIVERSITY OF PITTSBURGH
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.
Washington, DC: National Academy Press
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?
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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?
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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 the
Standards for Mathematical Content and Standards
for Mathematical Practice?
© 2013 UNIVERSITY OF PITTSBURGH
Functions and Modeling
In modeling situations, knowledge of the context and
statistics are sometimes used together to find algebraic
expressions that best fit an observed relationship between
quantities. Then the algebraic expressions can be used to
interpolate (i.e., approximate or predict function values
between and among the collected data values) and to
extrapolate (i.e., to approximate or predict function values
beyond the collected data values). One must always ask
whether such approximations are reasonable in the context.
http://commoncoretools.wordpress.com/2012/04/ccss_progression_functions_2012_12 04.bis.pdf, pg. 4
1. Buddy Bags
Creating Equations★
(A-CED)
Create equations that describe numbers or relationships
A-CED.A.2 Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels and
scales.
Interpreting Functions
(F-IF)
Interpret functions that arise in applications in terms of the context
F-IF.B.4
F-IF.B.5
★
For a function that models a relationship between two quantities,
interpret key features of graphs and tables in terms of the quantities,
and sketch graphs showing key features given a verbal description of
the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.★
Relate the domain of a function to its graph and, where applicable, to
the quantitative relationship it describes. For example, if the function
h(n) gives the number of person-hours it takes to assemble n engines in
a factory, then the positive integers would be an appropriate domain for
the function. ★
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. Disc Jockey Decisions
Creating Equations★
(A-CED)
Create equations that describe numbers or relationships
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 simple rational and exponential
functions.
A.CED.A.2 Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate
axes with labels and scales.
Reasoning with Equations and Inequalities
(A-REI)
Solve equations and inequalities in one variable
A.-REI.B.3 Solve linear equations and inequalities in one variable, including
equations with coefficients represented by letters.
★ 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
3. What’s the point?
Creating Equations★
(A-CED)
Create equations that describe numbers or relationships
A.CED.A.2
Create equations in two or more variables to represent
relationships between quantities; graph equations on
coordinate axes with labels and scales.
Reasoning with Equations and Inequalities
(A-REI)
Represent and solve equations and inequalities graphically
A.-REI.D.10
★
Understand that the graph of an equation in two variables is
the set of all its solutions plotted in the coordinate plane,
often forming a curve (which could be a line).
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, NGA Center/CCSSO, 2010
4. Paulie’s Pen
Creating Equations★
(A-CED)
Create equations that describe numbers or relationships
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 simple rational and exponential
functions.
A-CED.A.4
Rearrange formulas to highlight a quantity of interest, using
the same reasoning as in solving equations. For example,
rearrange Ohm’s law V = IR to highlight resistance R.
★ 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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