2.1

WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIPSUMMER INSTITUTE 2015

SESSION 2 • 16 JUNE 2015CONGRUENCE IN HIGH SCHOOL

2.2

TODAY’S AGENDA

Homework Review and discussion

Rigid motions and congruence (High School), Part I

The Case of Charlie Sanders

Dinner

Rigid motions and congruence (High School), Part II Reflecting on CCSSM standards aligned to Grade 8 congruence

Break

Student work analysis

Developing criteria for proof

Peer-teaching planning

Daily journal

Homework and closing remarks

2.3

LEARNING INTENTIONS AND SUCCESS CRITERIA

We are learning …

precise definitions of the basic rigid motions;

the CCSSM High School expectations for congruence

2.4

LEARNING INTENTIONS AND SUCCESS CRITERIA

We will be successful when we can:

use appropriate language to describe a basic rigid motion precisely;

explain the CCSSM High School congruence standards;

describe the progression in the CCSSM congruence standards from Grade 8 to High School.

2.5

ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

2.6

ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

Table discussion:

Compare your answers to last night’s “Extending the mathematics” prompt.

Identify common themes, as well as points of disagreement, in your responses to the “Reflection on teaching” prompt.

2.7

ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL), PART I

PRECISE DEFINITIONS OF ROTATIONS AND REFLECTIONS

ENGAGENY/COMMON CORE GRADE 10, LESSONS 12, 13 & 14

2.8

ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

Work with a partner to complete the task. You are allowed a protractor, compass, and straightedge.

Partner A: Without showing the card to your partner, describe how to draw the transformation indicated on the card. When you have finished, compare your partner’s drawing with the transformed image on your card. Did you describe the motion correctly?

Partner B: Your partner is going to describe a transformation to be performed on the figure on your card. Follow your partner’s instructions, and then compare the image of your transformation to the image on your partner’s card.

2.9

ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

What did you discover as you worked on the previous task?

Would your students reactions to the task be essentially the same as yours? Why or why not?

2.10

ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

Defining rotations

Read the definition of a rotation in Lesson 13 (page S.70)

2.11

ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

Finding angles and centers of rotations

Complete Exercises 1, 2 & 3 (pages S.72-73).

If you wish to use Geogebra, you may find this file useful:https://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day02/G10_M1_Lesson13_Ex1-3.ggb

2.12

ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

Exploration

Complete the Lesson 14 Exploratory Challenge (Page S.77)

If you wish to use Geogebra, you may find this file useful:https://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day02/G10_M1_Lesson14_Explore.ggb

What do you notice about these perpendicular bisectors?

2.13

ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

Defining reflections

Read the definition of a reflection in Lesson 14 (page S.78)

2.14

ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

Finding lines of reflection

Complete Examples 2, 3 & 4 (pages S.79 & S.80).

If you wish to use Geogebra, you may find these files useful:

https://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day02/G10_M1_Lesson14_Examples2-3.ggb

https://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day02/G10_M1_Lesson14_Example4.ggb

2.15

ACTIVITY 3 DEVELOPING IN STUDENTS A NEED FOR PROOF

THE CASE OF CHARLIE SANDERS

CASES OF REASONING AND PROVING IN SECONDARY MATHEMATICS

2.16

ACTIVITY 3 THE CASE OF CHARLIE SANDERS

With your small group, discuss what you noticed about the opportunities Charlie’s students had to engage in reasoning-and-proving.

In what ways were Charlie’s and Kathy’s implementations of the task sequence similar and in what ways were they different? Specifically, which of the Principles to Actions Mathematics Teaching Practices

did you notice in each classroom?

Discuss with your group, and record your most important similarity and difference on the whiteboards.

2.17

ACTIVITY 3 THE CASE OF CHARLIE SANDERS

What is The Case of Charlie Sanders a case of?

Dinner

2.19

ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL), PART II

PRECISE DEFINITION OF TRANSLATION

ENGAGENY/COMMON CORE GRADE 8, LESSON 16

2.20

ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

Constructing parallel lines

Use Geogebra to complete the Lesson 16 Exploratory Challenge (page S.89).

Explain why this construction works.

2.21

ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL)

Defining translations

Read the definition of a translation in Lesson 16 (page S.90)

Complete Examples 2 & 3 (page S.91) You may use Geogebra to complete either of these examples, but you may not use the translation tool: mimic the straightedge and compass constructions.

2.22

Read MP6, the sixth 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 MP6?

Are there other standards for mathematical practice that were prominent as you and your groups worked on the tasks?

ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE

Reflecting on CCSSM standards alignment

2.23

How do you see the expectations for the standards for mathematical practice change between Grade 8 and High School?

ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE

Reflecting on CCSSM standards alignment

2.24

Read the High School Congruence standards from the CCSSM.

How do you see the expectations for Geometry content change between Grade 8 and High School?

ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE

Reflecting on CCSSM standards alignment

Break

2.26

ACTIVITY 5 STUDENT WORK ANALYSIS

2.27

ACTIVITY 5STUDENT WORK ANALYSIS

Consider the following:

Prove the conjecture that when you add any two odd numbers, your answer is always even.

2.28

ACTIVITY 5STUDENT WORK ANALYSIS

Imagine that the students in your class produced responses A-J to the “odd + odd = even” task.

Review the ten student responses individually and record whether or not each response qualifies as a proof & provide the rationale that led you to that conclusion.

Discuss your ratings and rationale with members of your group, come to a group consensus on which responses are and are not proofs and why, and record you group’s decision on the Proof Evaluation Chart on the whiteboard.

As you work, be able to describe the criteria you’re using to judge whether or not an argument is a proof.

2.29

ACTIVITY 6 DEVELOPING CRITERIA FOR PROOF

2.30

ACTIVITY 6DEVELOPING A CRITERIA FOR PROOF

Some ideas from you:

Justify a given statement using facts

Uses universally agreed-upon truths

Contains a series of logical steps

You have finished when you get to the statement you set out to prove

Simplifies complex ideas

There can be no counterexamplesThink about the criteria for judging a mathematical argument as a proof.

2.31

ACTIVITY 6DEVELOPING A CRITERIA FOR PROOF

An argument that counts as proof must meet the following criteria:

The argument must show that the conjecture or claim is (or is not) true for all cases.

The statements and definitions that are used in the argument must be ones that are true and accepted by the community because they have been previously justified.

The conclusion that is reached from the set of statements must follow logically from the argument made.

In addition, a valid proof may vary along the following dimensions: 

type of proof (e.g., demonstration, induction, counterexample)

form of the proof (e.g., two-column, paragraph, flow chart)

representation used (e.g., symbols, pictures, words)

explanatory power (e.g., how well the proof explains why the claim is true)

Variance on these dimensions, however, does not matter as long as the preceding three criteria for proof are met.

2.32

ACTIVITY 7 PEER TEACHING PLANNING

2.33

Write your name on three separate Post-it notes, label each Post-it note as your first, second or third choice, and place the notes underneath the appropriate lesson number. Remember, you choices are:

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.

ACTIVITY 7 PEER TEACHING PLANNING

2.34

ACTIVITY 8 DAILY JOURNAL

2.35

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

ACTIVITY 8 DAILY JOURNAL

2.36

Complete Problems 5 and 6 from the Grade 10 Module 1 Lesson 16 Problem Set in your notebook (page S.95)

Extending the mathematics:We have made implicit use of parallel and perpendicular lines throughout our discussion of rigid motions. Identify as many of these uses as you can (from both the Grade 8 and Grade 10 material).

Reflecting on teaching:Consider a typical class of high school Geometry 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