1. Study Group 2 – Geometry Welcome Back!
Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises.
Our Bridge to Practice through TNCore Training is linking our classroom instruction to the CCSS!

2. Part A From Bridge to Practice #1:
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.
Go over Bridge to Practice Part A form last Study Group over Module 1
Part A: Show the next two slides to review the content AND practice standards for the selected CRA tasks.
Discuss the EVIDENCE that participants found for the standards they selected.
Remind participants that it would help to have a copy of the math practice standards from Module 1 handy (they are on the next slide for reference).
There is also an additional Powerpoint of the Math Practice Standards with visuals and clarifications called “CCSS Math Practices” as a reference, but there is not enough time in this session to go through each slide and discuss that Powerpoint. Teachers may want to keep it to refer to as they grow more familiar with the developing these practices in their students.

3. 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.
Use this slide as a quick reference to discuss the math practices relative to the two CRA problems.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO

4. 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.
(SAY): In what way does the prompt for the identified item elicit a student response that will demonstrate what s/he knows about specific content within the standard? (See content standards, slide 42.)
What is it about the prompt for the identified item that will require students to use the standards for mathematical practice?
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Students need to abstract information from the context and work to determine slopes and/or distances. They must understand the context as asking for perimeter. They must also remember to return to the context to indicate what their calculations have determined.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Students will again need to attend to precision of language, as well as process, in this task.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.

5. 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.
(SAY): In what way does the prompt for the identified item elicit a student response that will demonstrate what s/he knows about specific content within the standard? (See content standards, slide 43.)
What is it about the prompt for the identified item that will require students to use the standards for mathematical practice?
Make sense of problems and persevere in solving them.
Students must persevere in finding points the distance from a point on a line segment. They must look for evidence that the triangles so formed are congruent and use that evidence to prove the triangles congruent.
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.
Students may see and use properties of a parallelogram, look for evidence of SSS, SAS or ASA, etc. and use the results to prove the triangles congruent.
Look for and express regularity in repeated reasoning.

6. Part B from Bridge to Practice #1:
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.
Go over Bridge to Practice Part B from the last Study Group over Module 1
Lead a brief discussion over 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.

7. Part C From Bridge to Practice #1:
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.
Go over Bridge to Practice Part C from the last Study Group over Module 1
The remaining part of this session will be spent going through the following slides and analyzing the individual solutions each group member “brought to the table” for the instructional task.

8. Supporting Rigorous Mathematics Teaching and Learning
Engaging In and Analyzing Teaching and Learning through an Instructional Task
Tennessee Department of Education
High School Mathematics
Geometry
Overview of the Module:
Participants will consider what instruction that is aligned with the CCSS sounds like and looks like. We will engage in a lesson as adult learners. We will not pretend we are students or think about how students will respond. Instead we will engage in the lesson as adult learners. Our goal is to deepen our understanding of the standards and to make sense of the use of models when working with the concept.
No Prior Knowledge Necessary.
Materials:
Slides with note pages
Mathematics Common Core State Standards (CSSS) (the Standards for Mathematical Practice and the grade-level Standards for Mathematical Content)
Participant handouts (including Notes and Bridge to Practice #2)
Chart paper and markers
Graph paper
Rulers (optional)

9. Rationale
By engaging in an instructional task, teachers will have the opportunity to consider the potential of the task and engagement in the task for helping learners develop the facility for expressing a relationship between quantities in different representational forms, and for making connections between those forms.
(SAY) The CCSS include standards that focus on understanding of mathematical concepts AND the development of skills. We will engage in the lesson with the goal of deepening our understanding of concepts related to the task.

10. What is the difference between the following types of tasks?
Question to Consider…
What is the difference between the following types of tasks?
instructional task
assessment task
Ask participants if they have ever considered this question. Give them a minute to consider it and ask them to briefly share their thoughts.

11. Taken from TNCore’s FAQ Document:
Allow participants to read about the two types of tasks. Discuss a summary of the differences.
(ASK): What type of tasks are the CRAs we worked in our first study group? (Assessment tasks; however, they could be modified to be more open-ended to allow for more solutions paths and discussion)
(ASK): What type of task is The Building a New Playground Task from the Bridge to Practice? (Instructional task)

12. Session Goals Participants will:
develop a shared understanding of teaching and learning through an instructional task; and
deepen content and pedagogical knowledge of mathematics as it relates to the Common Core State Standards (CCSS) for Mathematics.
(This will be completed as the Bridge to Practice)
Directions:
Read the session goals.

13. Overview of Activities
Participants will:
engage in a lesson; and
reflect on learning in relationship to the CCSS.
(This will be completed as the Bridge to Practice #2)
(SAY) We will engage in a task, and then step out and reflect on our engagement in the task. We will consider how our learning was supported, and which standards we had opportunities to think about and use when figuring out the solution path.
We will engage with the task for the sake of our thinking and learning about the mathematics.
Facilitator Information:
If participants start to describe how their students would do the task or how their students think about the mathematics, remind them that for now we are focusing on our thinking and understanding of the task, and the underlying mathematics. Our goal in this module is to deepen our understanding of the standards and to make sense of the use of the mathematical practices when working with the concept(s).

14. Looking Over the Standards
Briefly look over the focus cluster standards.
We will return to the standards at the end of the lesson and consider:
What focus cluster standards were addressed in the lesson?
What gets “counted” as learning?
(SAY) Take a look at the focus cluster standards (on pages 11–12) in the handout. They are also on slides for reference, but you don’t need to select which standards this task addresses now because the Bridge to Practice will be over aligning the content standards with this particular task.

15. Building a New Playground Task
The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown.
(SAY): You already solved this task privately and now we are going to look at HOW to facilitate an instructional task by analyzing our solutions in small group and then whole group discussion.

16. The Structures and Routines of a Lesson
MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on:
Different solution paths to the
same task
Different representations
Errors
Misconceptions
Set Up of the Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/Small Group Problem Solving
Generate and Compare Solutions
Assess and Advance Student Learning
SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification.
REPEAT THE CYCLE FOR EACH SOLUTION PATH
COMPARE: Students discuss similarities and difference between solution paths.
FOCUS: Discuss the meaning of mathematical ideas in each representation
REFLECT: By engaging students in a quick write or a discussion of the process.
Share, Discuss, and Analyze Phase of the Lesson
1. Share and Model
2. Compare Solutions
3. Focus the Discussion on
Key Mathematical Ideas
4. Engage in a Quick Write
This slide is a model of how the Structures and Routines of a lesson should unfold as teachers facilitate an Instructional Task.
(SAY) This is how we will engage together. Structures and routines are patterned ways of working that help students know what to expect.
When we engage in lessons, we first set up the task. This usually takes a few minutes. Be careful not to provide students with too much scaffolding so they can develop their own ideas about how to solve it.
Then, you will have approximately 5 minutes of private time to solve the task independently. It is VERY important to give students this time prior to breaking up into groups so they can process the problem for themselves.
(For the purpose of our training, we completed the private think time as our Bridge to Practice)
Next, you will work in small groups for about 15 minutes. While you are working, I will circulate asking assessing and advancing questions. During this time, I will be looking for a variety of solution paths to have shared with the whole group. I may be asking you to write your method on chart paper.
Finally, we will engage in a group discussion of the different solution paths and make connections between the paths to arrive at the essential understandings of the standards related to this task.

17. Solve the Task (Private Think Time and Small Group Time)
Work privately on the Building a New Playground Task (This should have been completed as the Bridge to Practice prior to this session)
Work with others at your table. Compare your solution paths. If everyone used the same method to solve the task, see if you can come up with a different way.
Directions:
Read through the directions on the slide.
Focus participants on the ways in which they can find a location for the new playground.
Remind them that the goal is not to “know” and name the answer, but to examine alternate ways to find points that meet the conditions and to explain their reasoning about the results.

18. Expectations for Group Discussion
Solution paths will be shared.
Listen with the goals of:
putting the ideas into your own words;
adding on to the ideas of others;
making connections between solution paths; and
asking questions about the ideas shared.
The goal is to understand the mathematics and to make connections among the various solution paths.
Directions:
Use the information on the slide to describe the Share, Discuss, and Analyze Phase of the lesson.

19. Building a New Playground Task
The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown.
Facilitator Information:
This slide is here for you to reference during the group discussion.

20. Building a New Playground
PART A 
Determine at least three possible locations for the park that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools.
Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture.
PART B
The City Planning Commission is planning to build a third elementary school located at (8, -6) on the coordinate grid. Determine a location for the park that is equidistant from all three schools. Explain how you know that all three schools are equidistant from the park.
Describe a strategy for determining a point equidistant from any three points.
Facilitator Information:
This slide is here for you to reference during the group discussion.

21. Discuss the Task (Whole Group Discussion)
What patterns did you notice about the set of points that are equidistant from points A and B? What name can we give to that set of points?
Can we prove that all points in that set of points are equidistant from points A and B?
Have we shown that all the points that are equidistant from points A and B fall on that same set of points? Can we be sure that there are no other such points not on that set of points?
Directions:
The following sequence of mathematical concepts and questions can be used to guide your facilitation of the Share, Discuss, and Analyze Phase.
Teacher Questions Related to Essential Understandings  
The set of points that are equidistant from two points A and B lie on the perpendicular bisector of line segment AB, because every point on the perpendicular bisector can be used to construct two triangles that are congruent by reflection and/or Side-Angle-Side; corresponding parts of congruent triangles are congruent.
Intuitively determines that the point lies on the perpendicular bisector without supporting the conjecture with a proof.
Tell us about how the points you chose and why you believe they must lie on this single line.
How does the distance formula support your conjecture?
Can you prove that this will always be the case?
Uses SAS triangle congruence to determine that the equidistant points lie on the perpendicular bisector.
Tell us about how you used triangles to solve this problem.
Can you tell us about the sides and the angles of these triangles? Which corresponding sides and angles are congruent?
How did you use SAS? Can you use any other triangle congruence shortcuts?
Summary
What do we know about a set of points that are equidistant? Explain.
Why is this relationship true?
How does it relate to other topics that we’ve studied?
How many points are there? What is the relationship between all of the points?

22. Reflecting on Our Learning
What supported your learning?
Which of the supports listed will EL students benefit from during instruction?
Directions:
Chart responses.
Probing Facilitator Questions and Possible Responses:
What supported your learning?
Private think time
Small group problem solving time
The whole group discussion
The design of the task—It was open and we were able to come up with our own ways of thinking about the problem.
The Turn and Talk time—Because sometimes a question was hard to think about and I needed the time to think.
Did the talk help anyone else? What about the talk? Did anyone else find it helpful to hear others repeat ideas and to add on to other’s ideas?
Which of these supports would be helpful to English learners?
All of the things that supported our learning are good for all learners.
ELs will especially benefit from the structures built in that allow them to talk to their peers. Multiple representations are also helpful.
Cognates would be helpful if any apply to the problem. Cognates are the use of English words that have a word in Spanish that sounds similar. ELs might also benefit from hearing the problem before other students hear it. Teaching the vocabulary words in advance of the problem or having picture available would also be helpful.

23. Linking to Research/Literature Connections between Representations
Pictures
Written
Symbols
Manipulative
Models
Real-world
Situations
Oral
Language
(Say) Many of you moved between representations when solving and discussing the solution paths to the task.
Research has shown that some of the better problem solvers are those who, when they are struggling to figure out a problem, have the resources or know how to use different representations in order to solve a problem. If this is true, then what are the implications for instruction in our classrooms?
Adapted from Lesh, Post, & Behr, 1987

24. Five Different Representations of a Function
Language
Table
Context
Graph
Equation
(SAY) This diagram varies slightly from the one shown on the earlier slide. This one includes equations and graphs, which are more appropriate for middle school and high school students.
Both diagrams are included so that we can consider how students can benefit from looking at multiple representations of function relationships at all levels of mathematics learning.
Van De Walle, 2004, p. 440

25. 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.
Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO

26. 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.
Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO

27. 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.
Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2
Common Core State Standards, 2010, p. 77, NGA Center/CCSSO

28. 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.
Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2
Common Core State Standards, 2010, p. 78, NGA Center/CCSSO

29. Bridge to Practice #2: Time to Reflect on Our Learning
1. Using the Building a New Playground Task:
a. Choose the Content Standards from pages of the handout that this task addresses and find evidence to support them.
Choose the Practice Standards students will have the opportunity to use while solving this task and find evidence to support them.
Using the quotes on the next page, Write a few sentences to summarize what Tharp and Gallimore are saying about the learning process.
Read the given Essential Understandings. Explain why I need to know this level of detail about coordinate geometry to determine if a student understands the structure behind relationships.
Bridge to Practice #2: Make sure participants have a copy of the Bridge to Practice #2 handout that slides explain.

30. Research Connection: Findings by Tharp and Gallimore
For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.”
They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support.
For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that happen in conversation.
Bridge to Practice #2: 2) Read the Tharp and Gallimore Quotes. Write a few sentences to summarize what Tharp and Gallimore are saying about the learning process.
Tharp & Gallimore, 1991

31. Underlying Mathematical Ideas Related to the Lesson (Essential Understandings)
Coordinate Geometry can be used to form and test conjectures about geometric properties of lines, angles and assorted polygons.
Coordinate Geometry can be used to prove geometric theorems by replacing specific coordinates with variables, thereby showing that a relationship remains true regardless of the coordinates.
The set of points that are equidistant from two points A and B lie on the perpendicular bisector of line segment AB, because every point on the perpendicular bisector can be used to construct two triangles that are congruent by reflection and/or Side-Angle-Side; corresponding parts of congruent triangles are congruent.
It is sometimes necessary to prove both 'If A, then B' and 'If B, then A' in order to fully prove a theorem; this situation is referred to as an "if and only if" situation; notations for such situations include <=> and iff.
(Bridge to Practice #2: 3) Explain why I need to know this level of detail about coordinate geometry to determine if a student understands the structure behind relationships.