Supporting Rigorous Mathematics
Teaching and Learning
Engaging In and Analyzing Teaching and
Learning
Tennessee Department of Education
High School Mathematics
Geometry
© 2013 UNIVERSITY OF PITTSBURGH
Rationale
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.
Common Core State Standards for Mathematics, 2010
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.
© 2013 UNIVERSITY OF PITTSBURGH
Overview of Activities
Participants will:
• engage in a lesson; and
• reflect on learning in relationship to the CCSS.
© 2013 UNIVERSITY OF PITTSBURGH
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?
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
Building a New Playground
PART 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 B
3. 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.
© 2013 UNIVERSITY OF PITTSBURGH
The Structures and Routines of a Lesson
Set Up of the Task
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
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
© 2013 UNIVERSITY OF PITTSBURGH
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.
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.
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
Building a New Playground
PART 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 B
3. 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.
© 2013 UNIVERSITY OF PITTSBURGH
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?
© 2013 UNIVERSITY OF PITTSBURGH
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?
© 2013 UNIVERSITY OF PITTSBURGH
Linking to Research/Literature
Connections between Representations
Pictures
Manipulative
Models
Written
Symbols
Real-world
Situations
Oral
Language
Adapted from Lesh, Post, & Behr, 1987
Five Different Representations of a Function
Language
Context
Table
Graph
Van De Walle, 2004, p. 440
Equation
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?
© 2013 UNIVERSITY OF PITTSBURGH
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.
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
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.
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
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.
Common Core State Standards, 2010, p. 77, NGA Center/CCSSO
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
G-GPE.B.5
G-GPE.B.6
G-GPE.B.7
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).
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).
Find the point on a directed line segment between two given points
that partitions the segment in a given ratio.
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.
Common Core State Standards, 2010, p. 78, NGA Center/CCSSO
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?
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
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