Study Group 2 – Geometry

1. Study Group 2 – Geometry Welcome Back! Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises.

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.

3. The CCSS for Mathematical Practice Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO • 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.

4. 3. Lucio’sRide • When placed on a grid where each unit represents one mile, State Highway 111 runs along the line x + 3, and State Highway 213 runs along the linex - . • 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.

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

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.

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.

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

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.

10. Question to Consider… What is the difference between the following types of tasks? instructional task assessment task

11. Taken from TNCore’s FAQ Document:

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)

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)

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?

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.

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

17. Solve the Task(Private Think Time and Small Group Time) • Work privately on theBuilding 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.

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.

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.

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.

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?

22. Reflecting on Our Learning What supported your learning? Which of the supports listed will EL students benefit from during instruction?

23. Pictures Manipulative Models Written Symbols Real-world Situations Oral Language Linking to Research/LiteratureConnections between Representations Adapted from Lesh, Post, & Behr, 1987

24. Language Context Table Graph Equation Five Different Representations of a Function Van De Walle, 2004, p. 440

25. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry Common Core State Standards, 2010, p. 76, NGA Center/CCSSO

26. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry Common Core State Standards, 2010, p. 76, NGA Center/CCSSO

27. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry Common Core State Standards, 2010, p. 77, NGA Center/CCSSO

28. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry 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 11-12 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.

30. Research Connection: Findings by Tharp and Gallimore Tharp & Gallimore, 1991 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.

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.