Study Group 2 – Geometry
Welcome Back!
Let’s spend some quality time discussing what we learned
from our Bridge to Practice exercises.
© 2013 UNIVERSITY OF PITTSBURGH
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.
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The CCSS for Mathematical Practice
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, NGA Center/CCSSO
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3
3. Lucio’s Ride
When placed on a grid where each unit represents one mile, State Highway
3
111 runs along the line 𝑦 = x + 3, and State Highway 213 runs along the
line 𝑦 =
3
x
4
-
13
.
4
4
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).
a. Is the quadrilateral formed by connecting the
four locations a square? Explain why or why
not. Use slopes as part of the explanation.
b. 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.
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4. Congruent Triangles
a. Locate and label point M on
2
𝑆𝑈 such that it is of the
5
distance from point S to
point U. Locate and label
point T on 𝑆𝑁 such that it is
2
of the distance from point
5
S to point N. Locate and
label point Q on 𝑁𝑈 such
2
that it is of the distance
5
from point N to point U.
b. Prove triangles TNQ and
QMT are congruent.
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
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
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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.
Question to Consider…
What is the difference between the
following types of tasks?
• instructional task
• assessment task
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Taken from TNCore’s FAQ Document:
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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)
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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)
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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?
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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.
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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
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 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.
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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.
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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?
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Reflecting on Our Learning
• What supported your learning?
• Which of the supports listed will
EL students benefit from during
instruction?
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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
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.
© 2013 UNIVERSITY OF PITTSBURGH
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.
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.
© 2013 UNIVERSITY OF PITTSBURGH
G-CO.C.11
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.
© 2013 UNIVERSITY OF PITTSBURGH
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,UNIVERSITY
each standard
in that
© 2013
OF PITTSBURGH
domain is a modeling standard.
Common Core State Standards, 2010, p. 78, NGA Center/CCSSO
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.
b. Choose the Practice Standards students will have the opportunity to use
while solving this task and find evidence to support them.
2. Using the quotes on the next page, Write a few sentences to
summarize what Tharp and Gallimore are saying about the learning
process.
3. 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.
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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.
Tharp & Gallimore, 1991
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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.
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