Supporting Rigorous Mathematics Teaching and Learning

Supporting Rigorous Mathematics Teaching and Learning Engaging In and Analyzing Teaching and Learning Tennessee Department of Education High School Mathematics Geometry

Rationale Common Core State Standards for Mathematics, 2010 Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true….…Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. By engaging in a 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.

Session Goals Participants will: • develop a shared understanding of teaching and learning; and • deepen content and pedagogical knowledge of mathematics as it relates to the Common Core State Standards (CCSS) for Mathematics.

Overview of Activities Participants will: • engage in a lesson; and • reflect on learning in relationship to the CCSS.

Looking Over the Standards • Look over the focus cluster standards. • Briefly Turn and Talk with a partner about the meaning of the 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?

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.

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.

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

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

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.

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.

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.

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?

Reflecting on Our Learning What supported your learning? Which of the supports listed will EL students benefit from during instruction? Which CCSS for Mathematical Content did we discuss? Which CCSS for Mathematical Practice did you use when solving the task?

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

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

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

Which Standards for Mathematical Practice made it possible for us to learn? Common Core State Standards for Mathematics, 2010 • 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.

Research Connection: Findings from 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.

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.

Examples of Key Advances from Previous Grades or Courses – Geometry PARCC Model Content Frameworks for Mathematics, October 2011, pp. 53-54 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.

Supporting Rigorous Mathematics Teaching and Learning