Supporting Rigorous Mathematics
Teaching and Learning
Deepening Our Understanding of CCSS Via
A Constructed Response Assessment
Tennessee Department of Education
High School Mathematics
Geometry
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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.
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Overview of Activities
Participants will:
• analyze Constructed Response Assessments
(CRAs) in order to determine the way the
assessments 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.
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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
Geometry
 Understand congruence in terms of rigid motions.
 Prove geometric theorems.
 Define trigonometric ratios and solve problems
involving right triangles.
 Use coordinates to prove simple geometric
theorems algebraically.
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Analyzing Assessment Items
(Private Think Time)
Four assessment items have been provided:
 Park City Task
 Getting in Shape Task
 Lucio’s Ride Task
 Congruent Triangles Task
For each assessment item:
• solve the assessment item; and
• make connections between the standard(s) and the
assessment item.
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1. Park City Task
Park City is laid out on a grid like the one below, where each
line represents a street in the city, and each unit on the grid
represents one mile. Four other streets in the city are
represented by .
a. Dionne claims that the figure formed
by 𝐹𝐴, 𝐴𝐸 , 𝐸𝐶 , and 𝐶𝐹 is a
parallelogram. Do you agree or
disagree with Dionne? Use
mathematical reasoning to explain
why or why not.
b. Triangle AFE encloses a park located
in the city. Describe, in words, two
methods that use information in the
diagram to determine the area of the
park.
c. Find the exact area of the park.
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2. Getting in Shape Task
Points A (12, 10), J (16, 18), and Q (28, 12) are plotted on
. coordinate plane below.
the
a. What are the coordinates of a
point M such that the quadrilateral
with vertices M, A, J, and Q is a
parallelogram, but not a
rectangle?
b. Prove that the quadrilateral with
vertices M, A, J and Q is a
parallelogram.
c. Prove that the quadrilateral with
vertices M, A, J and Q is not a
rectangle.
d. Determine the perimeter of your
parallelogram.
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3. Lucio’s Ride
When placed on a grid where each unit represents one mile, State Highway
3
111 runs along the line 𝑦 = x + 3, and State Highway 213 runs along the line
𝑦=
3
x
4
-
13
.
4
4
The following locations are represented by points on the grid:
• Lucio’s house is located at (–3, 1).
• His school is located at (–1, –4).
• A grocery store is located at (–4, 0).
• His friend’s house is located at (0, 3).
a. Is the quadrilateral formed by connecting the
four locations a square? Explain why or why
not. Use slopes as part of the explanation.
b. Lucio is planning to ride his bike tomorrow. In
the morning, he plans to ride his bike from his
house to school. After school, he will ride to the
grocery store and then to his friend’s house.
Next, he will ride his bike home. The four
locations are connected by roads. How far is
Lucio planning to ride his bike tomorrow if he
plans to take the shortest route? Support your
response by showing the calculations used to
determine your answer.
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4. Congruent Triangles
a. Locate and label point M on
2
𝑆𝑈 such that it is of the
5
distance from point S to
point U. Locate and label
point T on 𝑆𝑁 such that it is
2
of the distance from point
5
S to point N. Locate and
label point Q on 𝑁𝑈 such
2
that it is of the distance
5
from point N to point U.
b. Prove triangles TNQ and
QMT are congruent.
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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?
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The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Congruence
(G-CO)
Understand congruence in terms of rigid motions
G-CO.B.6 Use geometric descriptions of rigid motions to transform figures
and to predict the effect of a given rigid motion on a given
figure; given two figures, use the definition of congruence in
terms of rigid motions to decide if they are congruent.
G-CO.B.7 Use the definition of congruence in terms of rigid motions to
show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles
are congruent.
G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS,
and SSS) follow from the definition of congruence in terms of
rigid motions.
Common Core State Standards, 2010
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Congruence
(G-CO)
Prove geometric theorems
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 segment are exactly those
equidistant from the segment’s endpoints.
G-CO.C.10 Prove theorems about triangles. Theorems include: measures
of interior angles of a triangle sum to 180°; base angles of
isosceles triangles are congruent; the segment joining
midpoints of two sides of a triangle is parallel to the third side
and half the length; the medians of a triangle meet at a point.
G-CO.C.11
Prove theorems about parallelograms. Theorems include:
opposite sides are congruent, opposite angles are congruent,
the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent
diagonals.
Common Core State Standards, 2010
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Similarity, Right Triangles, and Trigonometry (G-SRT)
Define trigonometric ratios and solve problems
involving right triangles
G-SRT.C.6 Understand that by similarity, side ratios in right
triangles are properties of the angles in the triangle,
leading to definitions of trigonometric ratios for acute
angles.
G-SRT.C.7 Explain and use the relationship between the sine and
cosine of complementary angles.
G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem
to solve right triangles in applied problems.★
★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 – Geometry
Expressing Geometric Properties with Equations (G-GPE)
Use coordinates to prove simple geometric theorems
algebraically
G-GPE.B.4
G-GPE.B.5
G-GPE.B.6
G-GPE.B.7
Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure
defined by four given points in the coordinate plane is a
rectangle; prove or disprove that the point (1, √3) lies on the
circle centered at the origin and containing the point (0, 2).
Prove the slope criteria for parallel and perpendicular lines and
use them to solve geometric problems (e.g., find the equation of
a line parallel or perpendicular to a given line that passes
through a given point).
Find the point on a directed line segment between two given
points that partitions the segment in a given ratio.
Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula.★
★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
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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.
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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.
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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
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.
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Assessing for Conceptual Understanding
The set of CRA items are designed to assess student
understanding of geometric properties.
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?
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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
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 the
Standards for Mathematical Practice?
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The CCSS for Mathematical Content K-8
Distribution of Domains K – 8
Domain
K
1
2
3
4
5
6
7
8
x
x
x
Ratios and Proportional
Relationships
x
x
The Number System
x
x
x
Expressions and Equations
x
x
x
Counting and Cardinality
x
Operations and Algebraic Thinking
x
x
x
x
x
x
Number and Operations in Base
Ten
x
x
x
x
x
x
x
x
x
Number and Operations - Fractions
Measurement and Data
x
x
x
x
x
x
Geometry
x
x
x
x
x
x
Functions
Statistics and Probability
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x
x
x
x
The CCSS for Mathematical Content −
High School
Geometry is one of 6 Conceptual Categories which
specify the mathematics that all students should study in
order to be college and career ready.
•
Number and Quantity
•
Algebra
•
Functions
•
Modeling
•
Geometry
•
Statistics and Probability
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Examples of Key Advances from
Previous Grades or Courses – Geometry
• Because concepts such as rotation, reflection, and
translation were treated in the grade 8 standards
mostly in the context of hands-on activities, and with
an emphasis on geometric intuition, high school
Geometry will put equal weight on precise
definitions.
• In grades K–8, students worked with a variety of
geometric measures (length, area, volume, angle,
surface area, and circumference). In high school
Geometry, students apply these component skills in
tandem with others in the course of modeling tasks
and other substantial applications (Mathematical
Practice #4).
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Examples of Key Advances from Previous
Grades or Courses – Geometry (continued)
The algebraic techniques developed in Algebra I can
be applied to study analytic geometry. Geometric
objects can be analyzed by the algebraic equations
that give rise to them. Some basic geometric
theorems in the Cartesian plane can be proven using
algebra.
PARCC Model Content Frameworks for Mathematics, October 2011, pp. 53-54
1. Park City
Expressing Geometric Properties with Equations (G-GPE)
Use coordinates to prove simple geometric theorems
algebraically
G-GPE.B.4 Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure
defined by four given points in the coordinate plane is a
rectangle; prove or disprove that the point (1, √3) lies on the
circle centered at the origin and containing the point (0, 2).
G-GPE.B.7 Use coordinates to compute perimeters of polygons and
areas of triangles and rectangles, e.g., using the distance
formula.★
★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. Moved By This Situation
Expressing Geometric Properties with Equations (G-GPE)
Use coordinates to prove simple geometric theorems
algebraically
G-GPE.B.4
G-GPE.B.7
Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure
defined by four given points in the coordinate plane is a
rectangle; prove or disprove that the point (1, √3) lies on
the circle centered at the origin and containing the point (0,
2).
Use coordinates to compute perimeters of polygons and
areas of triangles and rectangles, e.g., using the distance
formula. ★
★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. Lucio’s Ride
Expressing Geometric Properties with Equations (G-GPE)
Use coordinates to prove simple geometric theorems
algebraically
G-GPE.B.4 Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure
defined by four given points in the coordinate plane is a
rectangle; prove or disprove that the point (1, √3) lies on the
circle centered at the origin and containing the point (0, 2).
G-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines
and use them to solve geometric problems (e.g., find the
equation of a line parallel or perpendicular to a given line that
passes through a given point).
G-GPE.B.7 Use coordinates to compute perimeters of polygons and
areas of triangles and rectangles, e.g., using the distance
formula.★
★ 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. Congruent Triangles
Expressing Geometric Properties with Equations (G-GPE)
Use coordinates to prove simple geometric theorems
algebraically
G-GPE.B.4 Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a
figure defined by four given points in the coordinate
plane is a rectangle; prove or disprove that the point
(1, √3) lies on the circle centered at the origin and
containing the point (0, 2).
G-GPE.B.6 Find the point on a directed line segment between two
given points that partitions the segment in a given
ratio.
Common Core State Standards, 2010