Welcome to TNCore Training!
Introduction of 2013 CCSS Training
Tennessee Department of Education
High School Mathematics
Geometry
© 2013 UNIVERSITY OF PITTSBURGH
What this is / What it is not
What it is
What it is not
Peer led learning
Information updates from TDOE or
expert-delivered training
Content focused – we will dive deep
into understanding the expectations
Generic discussion of teaching
strategies
Focused on building our capacity
(knowledge and skill) as educators
Mandating implementation of a recipe
for instant success
Designed to meet participants at a
range of experience with Common
Core
Redundant of other TNCore trainings or – dependent on you having done
anything thus far
Focused on student achievement
Focused on compliance
Focused on your learning
Focused on preparing you to train others
© 2013 UNIVERSITY OF PITTSBURGH
Core Beliefs
Earning a living wage has
never demanded more
skills. This generation must
learn more than their parents
to do as well.
All children are capable of
learning and thinking at a high
level. Children in Tennessee
are as talented as any in the
country and often capable of
more than we expect.
Our current education results
pose a real threat to state and
national competitiveness and
security. Improving the skills of
our children is vital for the
future of Tennessee and
America.
Tennessee is on a mission to
become the fastest improving
state in the nation. Doing so
will require hard work and
significant learning for all. We
must learn to teach in ways we
were not taught ourselves.
There is no recipe that will
deliver a successful transition.
Preparing for Common Core
will demand effective
leadership focused on student
growth.
PARCC is coming in two years.
We need to use the transition
wisely to make sure our
students and our state are
ready.
© 2013 UNIVERSITY OF PITTSBURGH
Norms
• Keep students at the center of focus and decision-making
• Be present and engaged – limit distractions, if urgent matters
come up, step outside
• Monitor air time and share your voice - you’ll know which
applies to you!
• Challenge with respect – disagreement can be healthy, respect
all intentions
• Be solutions oriented – for the good of the group, look for the
possible
• Risk productive struggle - this is safe space to get out of your
comfort zone
• Balance urgency and patience - we need to see dramatic
change and change will happen over time
• Any other norms desired to facilitate your learning?
© 2013 UNIVERSITY OF PITTSBURGH
Emily Barton Video
© 2013 UNIVERSITY OF PITTSBURGH
Supporting Rigorous Mathematics
Teaching and Learning
Deepening Our Understanding of CCSS Via
A Constructed Response Assessment
Tennessee Department of Education
High School Mathematics
Geometry
© 2013 UNIVERSITY OF PITTSBURGH
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 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 the CCSS related to the tasks and the
implications for instruction and learning.
© 2013 UNIVERSITY OF PITTSBURGH
The Common Core State Standards
The standards consist of:
 The CCSS for Mathematical Content
 The CCSS for Mathematical Practice
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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.
© 2013 UNIVERSITY OF PITTSBURGH
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
© 2013 UNIVERSITY OF PITTSBURGH
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.
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.
© 2013 UNIVERSITY OF PITTSBURGH
Common Core State Standards, 2010
G-CO.C.11
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.
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
Common Core State Standards, 2010
Analyzing a
Constructed Response Assessment
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
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 𝐹𝐴, 𝐴𝐸, 𝐸𝐶 and 𝐶𝐹.
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.
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
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 ride 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.
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
Discussing Content Standards
(Small Group Time)
For each assessment item:
With your small group, find evidence in tasks 3 and 4
for the content standard(s) that will be assessed.
© 2013 UNIVERSITY OF PITTSBURGH
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
© 2013 UNIVERSITY OF PITTSBURGH
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
© 2013 UNIVERSITY OF PITTSBURGH
David Williams Video
© 2013 UNIVERSITY OF PITTSBURGH
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
26
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
27
Discussing Practice Standards
(Small Group Time)
Each person has a moment to share important
information about his/her assigned Mathematical
Practice.
© 2013 UNIVERSITY OF PITTSBURGH
28
Bridge to Practice:
Practice Standards
Choose the Practice Standards students will have the opportunity to use
while solving these tasks we have focused on and find evidence to support
them.
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?
Complete the Instructional Task
Work all of the instructional task “Building a New Playground” and be
prepared to talk about the task and the CCSSM Content and Practice
Standards associated with it.
© 2013 UNIVERSITY OF PITTSBURGH