1. Welcome to TNCore Training!
Introduction of 2013 CCSS Training
Tennessee Department of Education
High School Mathematics
Geometry
Overview of the Module:
(SAY): The purpose of this opening session is to familiarize teachers with both the Common Core State Standards (CCSS) for mathematical content in the selected domains of the CCSS for mathematical practice. We will carefully examine four Performance-Based Assessments (PBAs) that have the potential to assess both content and practices standards. In doing so, we will note the specific standards that each task has the potential to assess, and we will gain further clarity about the depth of understanding required by the standards. We will also identify characteristics of performance-based assessments. We will learn that the assessments measure several standards of mathematical content and several standards of mathematical practice.

2. What this 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
(SAY): As we begin this summer training it’s important to discuss both what this training is and is not. (You can read these aloud or have participants popcorn read in this format: The training is peer led learning, it is not information updates from TDOE or expert-delivered training.).
(DO) A 1 minute partner talk on any items that stand out as key takeaways and any items that are unexpected.

3. 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.
(SAY): This course is based on these six Core Beliefs.
Can I ask the group to popcorn read each of the core beliefs?

4. 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?
(SAY): Now that we’ve gone through the technical information of the week, let’s set some norms for how we will interact with one another. We’ve started with an initial list but will invite you to add any other norms that will make this a productive space.
Please take a minute to silently read through these on your own.
(Bring it back to whole group) – (SAY) Are there any norms that you believe will be particularly important or additional norms we want to add?

5. Emily Barton Video

6. Supporting Rigorous Mathematics Teaching and Learning
Deepening Our Understanding of CCSS Via A Constructed Response Assessment
Tennessee Department of Education
High School Mathematics
Geometry
Overview of the Module:
(SAY): The purpose of this opening session is to familiarize teachers with both the Common Core State Standards (CCSS) for mathematical content in the selected domains of the CCSS for mathematical practice. We will carefully examine four Performance-Based Assessments (PBAs) that have the potential to assess both content and practices standards. In doing so, we will note the specific standards that each task has the potential to assess, and we will gain further clarity about the depth of understanding required by the standards. We will also identify characteristics of performance-based assessments. We will learn that the assessments measure several standards of mathematical content and several standards of mathematical practice.

7. 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.
Directions:
Read the session goals.

8. 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.
Directions:
Give participants a sense of how they will spend the time working on this component.
(SAY) Today, we will focus on the use of PBAs that we might give to students after instruction on given concepts. Scored immediately, the PBAs allow teacher to gain insights into what students know or do not know about the concepts being assessed. As a result, the teacher can modify instruction and therefore, learning, in order to better meet the students’ needs.
We will develop an awareness of what a PBA looks like. We will discuss the characteristics of the assessment tasks and make connections between the tasks and the CCSS. In addition, we will consider the notion of conceptual understanding, and its relationship to the CCSS and PBAs. Finally, if we want our students to perform well on these assessments, we will consider what teaching and learning will look like in the classroom.
We will work individually, in small and in large groups over the course of this session, solving the assessment items, discussing the standards assessed by the items, and examining characteristics of PBA items.

9. The Common Core State Standards
The standards consist of:
The CCSS for Mathematical Content
The CCSS for Mathematical Practice
(SAY): As you already know, the CCSS consist of both standards for mathematical content and standards for mathematical practice.

10. 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.
Directions:
Note to participants that these are Tennessee’s Focus Clusters for the upcoming year.

11. 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.
Directions:
Note to participants that these are Tennessee’s Focus Clusters for the upcoming year.
Common Core State Standards, 2010

12. 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.
Directions:
Note to participants that these are Tennessee’s Focus Clusters for the upcoming year.
Common Core State Standards, 2010

13. 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.
Directions:
Note to participants that these are Tennessee’s Focus Clusters for the upcoming year.
Common Core State Standards, 2010

14. 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
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.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
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.
Directions:
Note to participants that these are Tennessee’s Focus Clusters for the upcoming year.
Common Core State Standards, 2010

15. Analyzing a Constructed Response Assessment
(SAY) Now we will turn our attention to the assessments.

16. 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.
Facilitator Note:
Refer participants’ attention to the pages with the tasks in their handouts.
Directions:
The Tennessee Department of Education has focused the work on selected clusters. For this first assessment, we will be focusing primarily on proof and proof through coordinate geometry.
While students will have engaged in some work with congruence prior to Geometry, it is in this course that serious thought is put into proof and the connections between algebra and geometry through the use of coordinate geometry. So let’s take a look at what that might mean.
Individually solve each task. As you do so, examine the Geometry standards to see which standards each item has the potential to assess..

17. 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 .
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.
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.
Find the exact area of the park.

18. 2. Getting in Shape Task
Points A (12, 10), J (16, 18), and Q (28, 12) are plotted on the coordinate plane below. .
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?  
Prove that the quadrilateral with vertices M, A, J and Q is a parallelogram.
Prove that the quadrilateral with vertices M, A, J and Q is not a rectangle.
Determine the perimeter of your parallelogram.

19. 3. Lucio’s Ride
When placed on a grid where each unit represents one mile, State Highway 111 runs along the line 𝑦= 3 4 x + 3, and State Highway 213 runs along the line 𝑦= 3 4 x
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).
Is the quadrilateral formed by connecting the four locations a square? Explain why or why not. Use slopes as part of the explanation.
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.

20. 4. Congruent Triangles
Locate and label point M on 𝑆𝑈 such that it is of the distance from point S to point U. Locate and label point T on 𝑆𝑁 such that it is of the distance from point S to point N. Locate and label point Q on 𝑁𝑈 such that it is of the distance from point N to point U.
Prove triangles TNQ and QMT are congruent.

21. 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.
(SAY) Determine the content standards related to each task.
Possible Responses:
(See the notes on slides for each task.)

22. 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
Facilitator Note:
These are the standards associated with the Lucio’s Ride Task.
Probing Facilitator Questions and Possible Responses:
Does this task meet standard G-GPE.B.5?
It meets a part of the standard; students must use criteria for parallel and perpendicular lines.
In this instance, it is unlikely that an assessment item will meet the other part. That part seems more like an instructional item.
This is the third time we have seen standards G-GPE.B.4 and G-GPE.B.7. What new information does this item give us about student understanding of these standards?
In this question, students must follow directions and create their own diagram to examine the situation.
They must make a connection between algebra and geometry in this task.
They must understand the context as asking for perimeter, as opposed to being directly asked for it.
Common Core State Standards, 2010

23. 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.
Facilitator Note:
These are the standards associated with the Congruent Triangles Task.
Probing Facilitator Questions and Possible Responses:
Does this task meet standard G-GPE.B.6?
Yes. Students must find the point on a directed line segment between two given points that partitions the segment in a ratio of 2 to 5 for each side of the triangle.
There is no other task that meets this standard in the assessment set. What confidence do we have that students understand this standard if they answer this correctly?
We know students can find a 2 to 5 ratio as opposed to a simpler one like 1 to 2. They do so three times. So we can have some confidence in their understanding of this standard.
Common Core State Standards, 2010

24. David Williams Video

25. Determining the Standards for Mathematical Practice Associated with the Constructed Response Assessment
(SAY): Now let’s turn our attention to the CCSS for Mathematical Practice.

26. 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.
Directions:
Read the directions on the slide.

27. The CCSS for Mathematical Practice
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.
Directions:
Show this list of standards for mathematical practice. Tell participants that they will be determining which of these standards for mathematical practice students will have an opportunity to use when solving the tasks.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO

28. Discussing Practice Standards (Small Group Time)
Each person has a moment to share important information about his/her assigned Mathematical Practice.
(SAY) You will have two minutes to share the gist of the mathematical practice that you studied. Others in your group should listen for similarities and differences among the practices.
After each person has shared the gist of their practice, discuss the similarities and differences among the practices.

29. 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.
Bridge to Practice
Participants should do three things before the next session:
Decide which practice standards students have the opportunity to use in the two tasks we focused on, and find evidence for them.
Think about what kind of instructional tasks/lessons need to be used and what teaching and learning needs to look like to prepare students for these assessments.
Complete the instructional task that goes with your subject area.