Supporting Rigorous Mathematics Teaching and Learning

© 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of EducationHigh School MathematicsGeometry

Engaging In and Analyzing Teaching and Learning

RationaleAsking 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.

Common Core State Standards for Mathematics, 2010


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 PlaygroundPART A 1. 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.

2. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture.

PART B3. 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.

4. Describe a strategy for determining a point equidistant from any three points.


The Structures and Routines of a Lesson

The Explore Phase/Private Work Time

Generate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions

2. Assess and Advance Student Learning

MONITOR: Teacher selects examples for the Share,

Discuss, and Analyze Phase based on:

• Different solution paths to the

same task

• Different representations

• Errors

• Misconceptions

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.

Set Up of the Task

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 the Building 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 PlaygroundPART A 1. 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.

2. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture.

PART B3. 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.

4. 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?

Linking to Research/LiteratureConnections between Representations

Pictures

WrittenSymbols

ManipulativeModels

Real-worldSituations

Oral Language

Adapted from Lesh, Post, & Behr, 1987

Five Different Representations of a Function

Language

TableContext

Graph Equation

Van De Walle, 2004, p. 440

The CCSS for Mathematical ContentCCSS 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, p. 76, NGA Center/CCSSO


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.



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.



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.


Which Standards for Mathematical Practice made it possible for us to learn?

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

Research Connection: Findings from 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.

Tharp & Gallimore, 1991


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

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.

PARCC Model Content Frameworks for Mathematics, October 2011, pp. 53-54

Supporting Rigorous Mathematics Teaching and Learning

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