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1.1
WELCOME TO COMMON CORE HIGH SCHOOL
MATHEMATICS LEADERSHIP
SUMMER INSTITUTE 2015
SESSION 1 • 15 JUNE 2015
GETTING THE BIG PICTURE &
CONGRUENCE IN GRADE 8
1.2
TODAY’S AGENDA
 Introductions, norms and administrative details
 Paperwork, surveys, and assessments
 Rigid motions and congruence (Grade 8)
 Dinner
 Applications of congruence (Grade 8)
 Reflecting on CCSSM standards aligned to Grade 8 congruence
 Break
 Considering the nature of proof
 Daily journal
 Homework and closing remarks
1.3
ACTIVITY 1
INTRODUCTIONS, NORMS AND ADMINISTRATIVE DETAILS
 Where do you teach and what do you teach?
 What are your learning goals for the summer?
 About mathematics content
 About mathematics teaching
1.4
ACTIVITY 1
INTRODUCTIONS, NORMS AND ADMINISTRATIVE DETAILS
Start on Time  End on Time
Silence cell phones.
Technology use for
learning only please!
No sidebar
conversations . . .
Name Tents
Attention signal
Raise hand!!
Food
• Administrative fee
Restrooms
1.5
ACTIVITY 2
MKT ASSESSMENT
GEOMETRY KNOWLEDGE ASSESSMENT
1.6
ACTIVITY 2
GEOMETRY KNOWLEDGE ASSESSMENT
MKT Assessment
Please give brief explanations for your rationale on
the multiple choice questions.
Link for online survey: http://bit.ly/15Geom
1.7
Standards for Mathematical Practice
Standards for Mathematics Content
• Make sense of problems & persevere in
solving them
• Reason abstractly & quantitatively
K–8 Standards
by Grade Level
• Construct viable arguments & critique the
reasoning of others
• Model with mathematics
• Use appropriate tools strategically
High School Standards
by Conceptual Categories
________________________
• Attend to precision
 Domains
• Look for & make use of structure
 Clusters
• Look for & express regularity in repeated
 Standards
reasoning
1.8
2. Reason abstractly and quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
6. Attend to precision.
1. Make sense of problems and
persevere in solving them.
STANDARDS FOR MATHEMATICAL PRACTICE
Reasoning and
Explaining
Modeling and
Using Tools
4. Model with mathematics.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
8. Look for and express regularity
in repeated reasoning.
William McCallum, The University of Arizona
Seeing Structure
and Generalizing
Which of these do you see as most
relevant to geometry?
1.9
1
2
3
4
5
6
7
8
HS
Counting
&
Cardinality
Algebra
Operations & Algebraic Thinking
Expressions and
Equations
Number & Operations in Base Ten
The Number System
Number &
Operations
Fractions
Ratios &
Proportional
Relationships
Measurement & Data
Functions
Statistics & Probability
Geometry
William McCallum, The University of Arizona
Number
and
Quantity
Modeling
K-8 Domains & HS
Conceptual Categories
K
1.10
AN OVERVIEW OF THE HIGH SCHOOL CONCEPTUAL
CATEGORIES
Algebra
Functions
Number &
Quantity
Geometry
Statistics & Probability
1.11
WHY THE ENGAGENY/COMMON CORE/
EUREKA MATH BOOKS?
 First curriculum designed from the ground up for Common Core
(not an existing curriculum “aligned” to the standards)
 Features tasks of high-cognitive demand that require students to think,
reason, explain, justify, and collaborate
 Contains substantial teacher implementation support resources:
lesson plans, notes on student thinking, assessments and rubrics
 Developed by an exceptional group of educators
 Source material is available free and could be integrated with existing
programs your district may have
1.12
LEARNING INTENTIONS AND SUCCESS CRITERIA
We are learning …
 to recognize and describe basic rigid motions;
 the definition of congruence in terms of basic rigid motions;
 the CCSSM Grade 8 expectations for congruence
1.13
LEARNING INTENTIONS AND SUCCESS CRITERIA
We will be successful when we can:
 use appropriate language to describe the rigid motion (or sequence of rigid motions) that take
one figure onto another;
 decide whether or not two figures are congruent;
 explain the CCSSM Grade 8 congruence standards.
1.14
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
DEFINITIONS AND PROPERTIES OF BASIC RIGID MOTIONS
ENGAGENY/COMMON CORE GRADE 8, LESSONS 1, 2, 4, 5, 10 & 11
1.15
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Describe, intuitively, what kind of
transformation will be required to
move the figure on the left to each of
the figures (1)-(3) on the right.
To help with this exercise, use a
transparency to copy the figure on the
left. Note: begin by moving the left
figure to each of the positions (1), (2)
and (3).
© 2014 Common Core, Inc.
1.16
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Given two segments AB and CD, which could be very far apart, how can we tell if they have
the same length without measuring them individually? Do you think they have the same
length? How can you check?
© 2014 Common Core, Inc.
1.17
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Vocabulary
 A transformation of the plane is a rule that assigns to each point of the plane exactly one
(unique) point.
 If we denote the transformation by F, we use F(P) to denote the unique point assigned to
P by F. (We will sometimes write P’ instead of F(P).)
 The point F(P) will be called the image of P by F.
 We also say F maps P to F(P).
 If given any two points P and Q, the distance between the images F(P) and F(P) is the same
as the distance between P and Q, then the transformation F preserves distance, or is
distance-preserving.
 A distance-preserving transformation is called a rigid motion (or an isometry); the name
suggests that it moves the points of the plane around in a rigid fashion.
1.18
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Turn and talk:
 What is a translation?
 A reflection?
 A rotation?
1.19
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Vocabulary
 A vector is a line segment with a specified direction
 If the segment is AB, and the specified direction is from A to B, we denote
the corresponding vector by
Vector
.
Vector
1.20
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Let’s play!
Use transparencies to model:
 A translation (along a given vector
)
 A reflection (across a given line)
 A rotation (of a given angle about a given point)
Note: translations, reflections, and rotations, are collectively known as basic
rigid motions.
1.21
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Properties of basic rigid motions
Explain why any basic rigid motion has the following three properties:
1. It takes lines to lines, segments to segments, rays to rays, and angles to
angles;
2. It preserves distances between points (and so also lengths of segments);
3. It preserves the degree measure of angles.
1.22
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Prove it!
Let P’ be the image of P by a reflection in line L. Use the properties of basic
rigid motions to explain why L must be the perpendicular bisector of the
segment PP’.
L
P’
.
.P
1.23
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Definitions
A congruence transformation (or just congruence, for short) is a sequence of
(one or more) basic rigid motions.
Two geometric figures are congruent if there is a congruence which takes one
of them onto the other.
1.24
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Properties of congruence transformations
Explain why any congruence transformation has the same three properties as
the basic rigid motions:
1. It takes lines to lines, segments to segments, rays to rays, and angles to
angles;
2. It preserves distances between points (and so also lengths of segments);
3. It preserves the degree measure of angles.
1.25
ACTIVITY 3
APPLICATIONS OF CONGRUENCE (GRADE 8)
Explore
 Complete Lesson 10, Exercises 1 through 4. (Pages S.51 &
S.52)
 Discuss Lesson 10 Problem Set 1, Problem 1 (Page S.54)
1.26
ACTIVITY 3
RIGID MOTIONS AND CONGRUENCE (GRADE 8)
Vocabulary
 A transformation of the plane is a rule that assigns to each point of the plane exactly one
(unique) point.
 If we denote the transformation by F, we use F(P) to denote the unique point assigned to
P by F. (We will sometimes write P’ instead of F(P).)
 The point F(P) will be called the image of P by F.
 We also say F maps P to F(P).
 If given any two points P and Q, the distance between the images F(P) and F(P) is the same
as the distance between P and Q, then the transformation F preserves distance, or is
distance-preserving.
 A distance-preserving transformation is called a rigid motion (or an isometry); the name
suggests that it moves the points of the plane around in a rigid fashion.
Dinner
1.28
ACTIVITY 4
WHAT IS PROOF?
PLEASE COMPLETE THE “WHAT IS PROOF?” QUESTIONS AS YOU EAT.
1.29
ACTIVITY 5
APPLICATIONS OF CONGRUENCE (GRADE 8)
ANGLES ASSOCIATED WITH PARALLEL LINES AND TRIANGLES
ENGAGENY/COMMON CORE GRADE 8, LESSONS 12 & 13
1.30
ACTIVITY 5
APPLICATIONS OF CONGRUENCE (GRADE 8)
Prove it!
Vertical angles, such as angle 1 and angle 3 in the diagram, are congruent, and so have equal
degree measure.
1
4
2
3
1.31
ACTIVITY 5
APPLICATIONS OF CONGRUENCE (GRADE 8)
Prove it!
If lines L1 and L2 are parallel, then corresponding angles, such as angles 1 and 5, are congruent,
and so have equal degree measure. Moreover, alternate interior angles, such as angles 4 and 6,
are congruent.
1
4
5
8
6
7
3
2
1.32
ACTIVITY 5
APPLICATIONS OF CONGRUENCE (GRADE 8)
Prove it!
The sum of the degree measures of the three angles in any triangle is 180O.
A
B
C
Hint. Start by drawing line CE parallel to line AB.
D
1.33
ACTIVITY 5
APPLICATIONS OF CONGRUENCE (GRADE 8)
Reflecting on CCSSM standards aligned to lesson 1
Read MP3, the third CCSSM standard for mathematical practice.
 Recalling that the standards for mathematical practice describe student behaviors,
how did you engage in this practice as you completed the lesson?
 What instructional moves or decisions did you see occurring during the lesson that
encouraged greater engagement in MP3?
 Are there other standards for mathematical practice that were prominent as you and
your groups worked on the tasks?
1.34
ACTIVITY 5
APPLICATIONS OF CONGRUENCE (GRADE 8)
CCSSM MP.2
MP.2 Reason abstractly and quantitatively
Mathematically proficient students make sense of quantities and
their relationships in problem situations. They bring two
complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given
situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without
necessarily attending to their referents—and the ability to
contextualize, to pause as needed during the manipulation process
in order to probe into the referents for the symbols involved.
Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units
involved; attending to the meaning of quantities, not just how to
compute them; and knowing and flexibly using different properties of
operations and objects.
EngageNY MP.2
MP.2 Reason abstractly and quantitatively.
This module is rich with notation that
requires students to decontextualize
and contextualize throughout.
Students work with figures and their
transformed images using symbolic
representations and need to attend to
the meaning of the symbolic notation
to contextualize problems. Students
use facts learned about rigid motions
in order to makes sense of problems
involving congruence.
1.35
ACTIVITY 5
APPLICATIONS OF CONGRUENCE (GRADE 8)
CCSSM MP.3
EngageNY MP.3
MP.3 Construct viable arguments and critique the reasoning of others.
MP.3 Construct viable arguments and critique the reasoning of
others.
Mathematically proficient students understand and use stated assumptions,
definitions, and previously established results in constructing arguments. They
make conjectures and build a logical progression of statements to explore the
truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their
conclusions, communicate them to others, and respond to the arguments of
others. They reason inductively about data, making plausible arguments that
take into account the context from which the data arose. Mathematically
proficient students are also able to compare the effectiveness of two plausible
arguments, distinguish correct logic or reasoning from that which is flawed,
and—if there is a flaw in an argument—explain what it is. Elementary students
can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even
though they are not generalized or made formal until later grades. Later,
students learn to determine domains to which an argument applies. Students
at all grades can listen or read the arguments of others, decide whether they
make sense, and ask useful questions to clarify or improve the arguments.
Throughout this module, students construct
arguments around the properties of rigid
motions. Students make assumptions about
parallel and perpendicular lines and use
properties of rigid motions to directly or
indirectly prove their assumptions. Students
use definitions to describe a sequence of
rigid motions to prove or disprove
congruence. Students build a logical
progression of statements to show
relationships between angles of parallel lines
cut by a transversal, the angle sum of
triangles, and properties of polygons like
rectangles and parallelograms.
Break
1.37
ACTIVITY 6
THE NATURE OF PROOF
1.38
ACTIVITY 6
THREE PROOF TASKS
 How many different 3-by-3 squares are in the 4-by-4 square shown?
 How many different 3-by-3 squares are there in a 5-by-5 square?
 How many different 3-by-3 squares are there in a 60-by-60 square?
Are you sure that your answer is correct? Why?
1.39
ACTIVITY 6
THREE PROOF TASKS
 Place different numbers of spots around a circle and join each pair of spots by
straight lines. Explore a possible relation between the number of spots and the
greatest number of non-overlapping regions into which the circle can be divided by
this means.
 When there are 15 spots around the circle, is there an easy way to tell for sure what
is the greatest number of non-overlapping regions into which the circle can be
divided?
(Geogebra or Geometer’s Sketchpad may be helpful here.)
1.40
ACTIVITY 6
THREE PROOF TASKS
 Discuss the student statement:
This problem teaches us that checking 5 cases is not enough to trust a
pattern in a problem. Next time I work with a pattern problem, I’ll check 20
cases to be sure.
1.41
ACTIVITY 6
THREE PROOF TASKS
 Does the expression 1 + 1141n2 (where n is a natural number) ever give a
square number?
No… until you get to 30,693,385,322,765,657,197,397,207
1.42
ACTIVITY 7
DAILY JOURNAL
1.43
ACTIVITY 7
DAILY JOURNAL
Take a few moments to reflect and write on today’s activities.
1.44
ACTIVITY 8
HOMEWORK AND CLOSING REMARKS
During Week 2 (June 29-July 2), together with a small group of your peers, you will teach one of
the following lessons (or some other one of your choice, with the approval of the course
facilitators):
 Grade 8. Module 2: Lesson 8 or Lesson 9.
 Grade 8, Module 3: Lesson 11 or Lesson 12.
 Grade 10, Module 1: Lessons 1 & 2, Lesson 3; Lesson 5; Lesson 9; or Lesson 10.
 Grade 10: Module 2: Lesson 7; Lesson 8; Lesson 9; Lesson 10; or Lesson 11.
You will be asked to select your preferred lesson tomorrow…
1.45
ACTIVITY 8
HOMEWORK AND CLOSING REMARKS
 Complete Problems 1, 2 and 3 from the Grade 8 Module 2 Lesson 13
Problem Set in your notebook (pages S.69-S.70)
 Extending the mathematics:
Can you find an example of a congruence that is not just a single basic rigid
motion? Can you prove that your example is not one of the basic types?
 Reflecting on teaching:
Consider a typical class of 8th grade students in your district. How would their
understanding of and experiences with congruence compare with what you
have see today?