1.1 welcome to common core high school mathematics leadership summer institute 2015 session 1 15...

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WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIPSUMMER INSTITUTE 2015

SESSION 1 • 15 JUNE 2015GETTING THE BIG PICTURE &CONGRUENCE IN GRADE 8

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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

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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

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Start on Time End on Time

Silence cell phones.Technology use for learning only please!

Name Tents

Attention signalRaise hand!!

RestroomsNo sidebarconversations . . .

Food• Administrative

fee

ACTIVITY 1 INTRODUCTIONS, NORMS AND ADMINISTRATIVE DETAILS

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ACTIVITY 2MKT ASSESSMENT

GEOMETRY KNOWLEDGE ASSESSMENT

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ACTIVITY 2GEOMETRY KNOWLEDGE ASSESSMENT

MKT Assessment

Please give brief explanations for your rationale on the multiple choice questions.

Link for online survey: http://bit.ly/15Geom

http://bit.ly/15Geom

http://bit.ly/15Geom

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• Make sense of problems & persevere in solving them

• Reason abstractly & quantitatively

• Construct viable arguments & critique the reasoning of others

• Model with mathematics

• Use appropriate tools strategically

• Attend to precision

• Look for & make use of structure

• Look for & express regularity in repeated reasoning

K–8 Standards

by Grade Level

High School Standards

by Conceptual Categories

________________________

Domains

Clusters

Standards

Standards for Mathematical Practice Standards for Mathematics Content

1.8STANDARDS FOR MATHEMATICAL PRACTICE1

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.2.Reason abstractly and quantitatively.

3.Construct viable arguments and critique the reasoning of others.

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.

Reasoning and Explaining

Modeling and Using Tools

Seeing Structure and Generalizing

William McCallum, The University of Arizona

Which of these do you see as most relevant to geometry?

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William McCallum, The University of Arizona

K 1 2 3 4 5 6 7 8 HS Counting

& Cardinality

Algebra

Number and

Quantity

Mo

de

lin

g

Operations & Algebraic Thinking

Expressions and

Equations

Number & Operations in Base Ten

The Number System

Number & Operations Fractions

Ratios & Proportional

Relationships

Functions

Measurement & Data

Statistics & Probability

Geometry

K-8

Dom

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HS

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Modeling

Functions

Number & Quantity

Statistics & Probability

Geometry

Algebra

AN OVERVIEW OF THE HIGH SCHOOL CONCEPTUAL CATEGORIES

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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

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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

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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.

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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

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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.

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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.

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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.

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Turn and talk:

What is a translation?

A reflection?

A rotation?

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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

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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.

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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.

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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’ .

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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.

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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.

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ACTIVITY 3APPLICATIONS 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)

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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

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ACTIVITY 4 WHAT IS PROOF?

PLEASE COMPLETE THE “WHAT IS PROOF?” QUESTIONS AS YOU EAT.

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ACTIVITY 5 APPLICATIONS OF CONGRUENCE (GRADE 8)

ANGLES ASSOCIATED WITH PARALLEL LINES AND TRIANGLES

ENGAGENY/COMMON CORE GRADE 8, LESSONS 12 & 13

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Prove it!

Vertical angles, such as angle 1 and angle 3 in the diagram, are congruent, and so have equal degree measure.

1 2

34

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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 2

34

56

78

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Prove it!The sum of the degree measures of the three angles in any triangle is 180O.

A

BC D

Hint. Start by drawing line CE parallel to line AB.

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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?


Reflecting on CCSSM standards aligned to lesson 1

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CCSSM MP.2MP.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.2MP.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.

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CCSSM MP.3MP.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.

EngageNY MP.3MP.3 Construct viable arguments and critique the

reasoning of others.

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.

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ACTIVITY 6 THE NATURE OF PROOF

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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?

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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.)

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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.

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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

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ACTIVITY 7 DAILY JOURNAL

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Take a few moments to reflect and write on today’s activities.

ACTIVITY 7 DAILY JOURNAL

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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…

ACTIVITY 8 HOMEWORK AND CLOSING REMARKS

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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?

ACTIVITY 8 HOMEWORK AND CLOSING REMARKS

1.1 welcome to common core high school mathematics leadership summer institute 2015 session 1 15...

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