DUE 6-13: Facilitators Guide Template - CC 6-12.docx

Module Focus: Grade8 – Module 2

Sequence of Sessions

Overarching Objectives of this November 2013 Network Team Institute

· Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate how these modules contribute to the accomplishment of the major work of the grade.

· Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.

· Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

· Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.

High-Level Purpose of this Session

· Implementation: Participants will be able to articulate and model the instructional approaches to teaching the content of the first half of the lessons.

· Standards alignment and focus: Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade.

· Coherence: Participants will be able to articulate connections from the content of previous grade levels to the content of this module.

Related Learning Experiences

· This session is part of a sequence of Module Focus sessions examining the Grade 7 curriculum, A Story of Ratios.

Key Points

· Experimental verifications of the basic rigid motions.

· Precision of language in descriptions of sequences.

· Immediate application of concept of congruence for understanding angle relationships.

· Application of congruence and angle relationships to explain a proof of the Pythagorean theorem.

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

· Understand the sequence of mathematical concepts within this module.

· Articulate and model the instructional approaches that support implementation of this module (both as classroom teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.

· Articulate the connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

· Articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.

Participants will be able to articulate the key points listed above.

Session Overview

Section

Time

Overview

Prepared Resources

Facilitator Preparation

Introduction to the Module

8 mins

Introduction to the instructional focus of Grade 8 Module 2 of A Story of Ratios.

· Grade 8 – Module 2

· Grade 8 – Module 2 PPT

Concept Development

108 mins

Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons.

· Grade 8 – Module 2

· Grade 8 – Module 2 PPT

· Grade 8 – Module 2 Lesson Notes

Review Grade 8 Module 2 Overview, Topic Openers, and Assessments.

Module Review

4 mins

Articulate the key points of this session.

Session Roadmap

Section: Introduction to the Module

Time: 20 minutes

[8 minutes] In this section, you will…

Introduce the instrutional focus of Grade 8 Module 2 of A Story of Ratios.

Materials used include:

Time

Slide #

Slide #/ Pic of Slide

Script/ Activity directions

GROUP

2

1.

Welcome! In this module focus session, we will examine Grade 8 – Module 2.

2

2.

Our objectives for this session are to:

· Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons.

· Identify the big idea within each topic in order to support instructional choices that achieve the lesson objectives while maintaining rigor within the curriculum.

.

2

3.

We will begin by exploring the module overview to understand the purpose of this module. Then we will dig in to the math of the module. We’ll lead you through the teaching sequence, one concept at a time. Along the way, we’ll also examine the other lesson components and how they function in collaboration with the concept development. Finally, we’ll take a look back at the module, reflecting on all the parts as one cohesive whole.

Let’s get started with the module overview.

2

4.

The second module in Grade 8 is called The Concept of Congruence. The module is allotted 25 instructional days. This module will be challenging for students and for teachers as it contains concepts that are typically reserved for HS Geometry or treated superficially in the past. Module 2 takes a hard look at the concept of congruence in terms of the basic rigid motions. No longer is the usual “same size, same shape” definition used as the true definition comes to life through the learning of translation, rotation, and reflection. The concept of congruence is applied to angle relationships of parallel lines, leading to an understanding of the triangle sum theorem. Two optional lessons are included in this module related to the Pythagorean theorem.

Section: Concept Development- Topic A

Time: 48 minutes

[108 minutes] In this section, you will…

Examine the conceptual understandings that are built in Grade 8 Module 2.

Materials used include:

PPT and Presenter Notes

Time

Slide #

Slide #/ Pic of Slide

Script/ Activity directions

GROUP

0

5.

1

6.

“These are the main points from the lesson. The goal is for students to develop an intuitive understanding of moving things around on the plane.”

3

7.

“One of the ways to develop that intuitive sense is with this exploratory challenge. Students complete the exploratory challenge with the instruction to:

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 that you are supposed to begin by moving the left figure to each of the locations in (1), (2), and (3).”

Ask participants to describe the moves necessary. Then show the answer.

1

8.

“By the end of the lesson, we expect students to know how to use the tools of the transparency and marker, as well as have an understanding about transformations that preserve distance, known as rigid motions, which is the focus for many of the lessons in this module.”

1

9.

“In Lesson 2, students learn about the rigid motion translation and its properties.”

2

10.

“Students verify experimentally the properties of translation, with an exercise like the one shown here.”

Have participants compete this exercise with the instruction:

“The diagram below shows figures and their images under a translation along HI. Use the original figures and the translated images to fill in missing labels for points and measures.”

1

11.

Ask for or present the answers.

1

12.

By the end of the lesson, we expect students to know what a translation is, how to perform one, and what “prime” notation is and how to use it.”

1

13.

In this lesson, students examine the effect of translation on lines.”

2

14.

“This first exercise is the basis for understanding the concept of the parallel postulate. Though not named, the idea that only one line exists through a given point parallel to a given line is used in this and the modules that follow.”

“Students complete exercises that require them to translate lines and make observations about the line and its image.”

1

15.

Answers to the first two exercises:

Top: There is only one line passing through a given point, that is parallel to the given line.

Bottom: Line L and its image coincide.

1

16.

Top: Line L and its image coincide.

Bottom: Line L and its image are parallel.

Understanding that translations of lines produce an image that is either the line itself or a line parallel to the given line rely on the work completed at the end of Lesson 2 about the translation of a point.

Note that references to “A above” and “B above” should be replaced by “Lesson 2” and that the exercise numbers referenced do not match. (Exercise 4 should be Exercise 2, Exercise 5 should be Exercise 3, and Exercise 6 should be Exercise 4.)

1

17.

“By the end of the lesson, we expect students to know what happens to lines when they’ve been translated. Note that there is a discrepancy in the summary.”

3

18.

“In Lesson 4, students learn about the rigid motion reflection and its properties.”

1

19.

“Note that we do not refer to this as the perpendicular bisector. Here we focus on the fact that the line of reflection is the bisector of the segment that connects a point and its image. Ask participants, How can we be sure that Line L bisects PP’? Answer: Reflections preserve segment length. Therefore the length of PO is the same as length P’O. Therefore Line L bisects PP’.”

1

20.

“By the end of the lesson, we expect students to know the properties of reflection, that a point on the line of reflection does not move, that a point off of the line, when connected to its’ image, is bisected by the line of reflection.”

3

21.

“In Lesson 5 students learn about rotation and its properties.”

1

22.

“Notice that in some cases we use positive and negative degrees of rotation. Positive degrees of rotation are always counterclockwise rotations. Think about how you measure using a protractor. The numbers start at 0 on the bottom right of the protractor and increase as you move counterclockwise. Angles and rotations are always measured this way. For that reason, a negative rotation is clockwise.”

Show participants on the document camera how to use a transparency to rotate in multiples of 90˚.

3

23.

By the end of the lesson, we expect students to know how to perform a rotation, know the properties of rotation, and know how to describe a rotation.”

1

24.

Lesson 6 focuses on rotations of 180 degrees on the coordinate plane. We don’t expect students to memorize rules related to rotation, but understand the properties of such a rotation so that the “rule” makes sense.”

2

25.

“What are the locations of the rotated points? How did you determine the coordinates?”

1

26.

Students will know that a point, its image and the center of rotation are collinear. Students can use their transparency to rotate the point, around the origin, and locate its image.”

1

27.

“By the end of the lesson, we expect students to know the effect that rotation has on points in the coordinate plane.”

1

28.

“This is the first lesson of Topic B. Students begin their work of sequencing rigid motions by sequencing translations. Mapping a figure back onto itself is the basis for understanding congruence in general.”

3

29.

Review the discussion points with the participants. State clearly that “The reason we want to discuss this is to lay the groundwork for mapping a figure onto itself and more specifically the concept of congruence. We begin by translating an object once, and then again, back to its original position. It should be obvious that the figure did not change under the translations.”

1

30.

“By the end of the lesson, we expect students to know that they can sequence translations. Further, that if a figure is moved, it can be moved back.”

1

31.

“This lesson continues with sequences of reflections and translations.”

2

32.

“In one of the discussions of this lesson students are asked if the order in which rigid motions are performed has an impact on the final location of the image. In the diagram, the final location of the ellipse after the reflection and translation is the lower red ellipse.”

2

33.

“In this diagram, the final location of the ellipse after the translation and the reflection is the red ellipse on the right. When students compare the two diagrams, is should be obvious that the order in which transformations are performed matters.”

1

34.

“By the end of the lesson, we expect students to know that they can sequence more than one kind of rigid motion and that the order of the sequence makes a difference in the location of the image.”

1

35.

“In this lessons students experiment with rotations around the same center and different centers.”

4

36.

“In Exploratory Challenge 1, students rotate a figure around the same center. In this challenge, students rotate around two different centers, then try to locate a center of rotation that would take the twice rotated image back to its original position. Students will not learn how to locate the center of rotation until grade 10 geometry. Until then, they can locate it experimentally using their transparency.”

1

37.

Here are the answers to the questions from the Exploratory Challenge:

 b) No, a single rotation around center D will not map △A'B'C' onto △ABC.

 

c) No, a single rotation around center E will not map △A'B'C' onto △ABC.

 

d) Yes, a rotation around center F, d degrees will map △A'B'C' onto △ABC.

“Students can only find the center F through trial and error at this point.” Demonstrate how to find the center of rotation using the document camera.

1

38.

“By the end of the lesson, we expect students to know that they can sequence rotations. Students know that the order in which rotations are performed matters when there are two different centers.”

1

39.

“Finally, students sequence all three rigid motions. At this point we expect students to be fairly proficient at performing the motions so the focus should be on describing the sequence with attention to the precision of the description.”

4

40.

Ask participants to describe the sequence for each scenario. Show the solution using the document camera. Hold participants accountable for using precise language in their descriptions. For example, instead of “rotate triangle A’B’C’” they should state “rotate triangle A’B’C’, d degrees, around center A”.

1

41.

“By the end of the lesson, we expect students to be able to describe a sequence of rigid motions that maps one figure onto another.”

1

42.

“In this lesson, students formalize their understanding of sequences of rigid motions as congruence.”

3

43.

“The point of this exercise is to show that congruence is transitive, but also to have students practice describing sequences and become familiar with the notation related to congruence.” If time, have participants provide descriptions of at least one sequence.

1

44.

“By the end of the lesson, we expect students to know that congruence is defined as mapping one figure onto another by a sequence of rigid motions.”

1

45.

“Lesson 12 expands students’ knowledge of angles to include those that are associated with parallel lines.”

4

46.

“This activity is preceded by one where the lines are not parallel. In the discussion about the exercise, the terms corresponding, alternate interior and alternate exterior angles are introduced to students. They are then practiced in the debrief of this exercise. Let’s look at the questions that follow this challenge.”

4

47.

Review the responses with the participants. “We base the understanding of the angles being congruent in basic rigid motions to reinforce the concept.”

1

48.

“By the end of the lesson, we expect students to know which angles associated with parallel lines are equal, and why.”

1

49.

“Lesson 13 exposes students to two proofs of the angle sum of a triangle. One of which we will look at more closely next.”

4

50.

“Students are asked the following guiding questions to help with the proof:

Name the triangle in the figure.

∆BCF

Name a straight angle that will be useful in proving that the sum of the interior angles of the triangle is 180˚.

∠GFE

As before, our goal is to show that the interior angles of the triangle are equal to the straight angle. Use what you learned from Exploratory Challenge 1 to show that interior angles of a triangle have a sum of 180˚.

 

Write your proof below.

The straight angle, ∠GFE is comprised of angles ∠GFB, ∠BFC, ∠EFC. Alternate interior angles of parallel lines are equal in measure, (alt. ∠s, AD∥CE). For that reason, ∠BCF=∠EFC and ∠CBF=∠GFB. Since ∠GFE is a straight angle, it is equal to 180˚. Then ∠GFE=∠GFB+∠BFC+∠EFC=180°. By substitution, ∠GFE=∠CBF+∠BFC+∠BCF=180°. Therefore, the sum of the interior angles of a triangle is 180. (∠ sum of △).“

1

51.

“By the end of the lesson, we expect students to know that the sum of the interior angles of a triangle is 180.”

1

52.

“Lesson 14 teaches students about exterior angles and their relationship to the remote interior angles of a triangle.”

4

53.

Ask participants to explain why the measure of an exterior angle is equal to the sum of the related remote interior angles. The reason is as follows:

Triangle XYZ has interior angles ∠XYZ, ∠YZX and ∠ZXY. The sum of those angles is 180˚. The straight angle ∠XYP also has a measure of 180˚ and is made up of angles ∠XYZ and ∠ZYP. Since the triangle and the straight angle have the same number of degrees, we can write the sum of their respective angles as an equality: ∠XYZ+∠YZX+∠ZXY=∠XYZ+ZYP.

Both the triangle and the straight angle share ∠XYZ. We can subtract the measure of that angle from the triangle and the straight angle. Then we have:∠YZX+∠ZXY=∠ZYP

where the angle ∠ZYP is the exterior angle and the angles ∠YZX and ∠ZXY are the related remote interior angles of the triangle. Therefore, the sum of the remote interior angles of a triangle are equal to the exterior angle.

1

54.

“By the end of the lesson, we expect students to be able to identify the exterior angles of a triangle, and determine their measure.”

1

55.

“Lesson 15 is the first proof that students see for the Pythagorean theorem. They are not expected to know one on their own until Module 7.”

3

56.

Ask the participants to answer the questions:

Looking at the outside square only, the square with side lengths (a+b), what is its area?

(a+b)^2=a^2 +2ab+b^2

Are the four triangles congruent? How do you know?

Yes. Use a transparency.

What is the area of one triangle? What is the total area of the four triangles?

½ ab, 2ab

3

57.

Ask participants to answer the questions. “The figure inside is a square because of what we know about congruent triangles, triangle sum and angles on a line. The angles of the figure in the center have no choice but to be 90˚.

The area is c^2.”

The answer to the last question is on the next slide.

1

58.

Show the solution and the end of the proof. Ask participants if they have any questions about the proof.

1

59.

“By the end of the lesson, we expect students to be able to apply the Pythagorean theorem to right triangles.”

1

60.

“In this lesson, students apply the Pythagorean theorem in real world and mathematical problems.”

1

61.

Students learn to find the distance between two points on the coordinate plane by drawing in the legs of a right triangle.”

1

62.

“By the end of the lesson, we expect students to apply the Pythagorean theorem to a variety of situations.”

Section: Module Review

Time: 8 minutes

[8 minutes] In this section, you will…

Faciliate as participants articulate the key points of this session and clarify as needed.

Materials used include:

Time

Slide #

Slide #/ Pic of Slide

Script/ Activity directions

GROUP

0

63.

2

64.

Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have?

Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

2

65.

Let’s review some key points of this session.

Use the following icons in the script to indicate different learning modes.

Video

Reflect on a prompt

Active learning

Turn and talk

Turnkey Materials Provided

· Grade 8 Module 2 PPT

· Grade 8 Module 2 Lesson Notes

Additional Suggested Resources

· A Story of Ratios Curriculum Overview

· CCSS Progressions Document: The Number System (6-8)