05.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SCHOOL YEAR SESSION 5 4 NOVEMBER 2015 SLICING AND DICING

05.2 TODAY’S AGENDA  Exploring the PRIME document  Cross sections of pyramids and cones (Eureka Math TM G10 M3 L7)  Break  “Big Ideas” discussion  Pre-observation tool  Modeling mentoring conversations  Homework and closing remarks

05.3 ACTIVITY 1 THE PRIME FRAMEWORK

05.4 ACTIVITY 1 THE PRIME FRAMEWORK With your shoulder partner:  Discuss your completion of the PRIME template (assigned for homework)  Complete the Moving to Action with Teacher Leadership form for your plan (with assistance from your partner)

05.5 ACTIVITY 1 THE PRIME FRAMEWORK Still with your shoulder partner:  Discuss bullet points you would like to add to each of the following posters:  Actions taken by teacher leaders  Actions outside of your control  Outcomes of a teacher leader  You should have (at least) one bullet point for each poster

05.6 ACTIVITY 1 THE PRIME FRAMEWORK With a new shoulder partner:  Discuss your completion of the self-assessment rubric (assigned for homework)  Summarize the primary component of the rubric related to your plan, and indicate specific actions you intend to take  Prepare a phrase, or a few words, describing key actions or activities that will advance your plan

05.7 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES EUREKA MATH TM GRADE 10 MODULE 3 LESSON 7

05.8 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Prelude  Individually, create a concept map around the idea of volume.  Chat briefly with your small groups about what you created.  We’ll bring this concept map back in the future.

05.9 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Warm-up: When a screen is placed 5 feet away from a projector, the picture on the screen measures 3 feet by 4 feet. What will the picture dimensions be if the screen is placed 10 feet away from the projector? 15 feet? 1 foot?

05.10 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES  Complete the Opening Exercise in Lesson 7

05.11 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Lesson goals  To understand the definition of a general cone, and that the cross sections of a cone are similar to the base  To show that if two cones have the same base area and the same height, then cross sections for the cones at the same distance from their respective vertices have equal areas As you engage, consider: what aspects of Big Ideas 3 and 4 do you notice in the lesson? Record tangible examples of each to share at the lesson’s conclusion.

05.12 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Turn and talk:  How would you define a general cylinder?  How would you define a general cone?  How would you define a cross section of a cylinder or a cone?

05.13 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Explore the Geogebra applets below. What do you notice?    (For both, you can view the worksheet to interact in the browser or download into the Geogebra app.)

05.14 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Definition of a general cylinder:  Let E and E’ be two parallel planes, let B be a region in the plane E, and let L be a line which intersects E and E’ but not B. At each point P of B, consider the segment parallel to L and intersecting E at a point P’. The union of all these segments is called a general cylinder with base B.

05.15 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Definition of a general cylinder:

05.16 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Examples of general cylinders:

05.17 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Definition of a general cone:  Let B be a region in a plane E, and let V be a point not on E. At each point P of B, consider the segment joining P to V. The union of all these segments is called a general cone with base B.

05.18 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Definition of a general cone:

05.19 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Definition of a cross section: A cross section of a general cylinder or cone is the intersection of the cylinder or cone with a plane parallel to the base.

05.20 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Cross sections of a general cone:

05.21 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Conjecture:  The cross sections of a general cone are similar to the base. We have already seen an informal argument for this conjecture, based on dilations in 3 dimensions (which we have not yet studied carefully). We will now look at a more precise argument, based on triangular pyramids.

05.22 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Exercise 1:

05.23 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Exercise 2:

05.24 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Theorem (General Cone Cross-section Theorem)  If two general cones have the same base area and the same height, then cross-sections for the general cones the same distance from the vertex have the same area.

05.25 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Warm-up: When a screen is placed 5 feet away from a projector, the picture on the screen measures 3 feet by 4 feet. What will the picture dimensions be if the screen is placed 10 feet away from the projector? 15 feet? 1 foot?

05.26 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Explore the Geogebra applets below. How do these illustrate aspects of the mathematics we just explored?    (For both, you can view the worksheet to interact in the browser or download into the Geogebra app.)

05.27 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Lesson goals  To understand the definition of a general cone, and that the cross sections of a cone are similar to the base  To show that if two cones have the same base area and the same height, then cross sections for the cones at the same distance from their respective vertices have equal areas

Break

05.29 ACTIVITY 3 “BIG IDEAS” DISCUSSION FROM NCTM ESSENTIAL UNDERSTANDINGS FOR GEOMETRY GRADES 9-12

05.30 ACTIVITY 3 “BIG IDEAS” DISCUSSION  Reflect on your own about how the focus on this Big Idea allowed you to make connections between this lesson and other Geometry concepts.

05.31 ACTIVITY 4 MODELING MENTORING CONVERSATIONS

05.32 ACTIVITY 4 MODELING MENTORING CONVERSATIONS  Five phases of an instructional conference:  Launching  Reflecting  Reinforcing  Refining  Reflecting

05.33 ACTIVITY 4 MODELING MENTORING CONVERSATIONS  As Mike conducts the conference, consider the following:  What relational skills do you notice being used in the conference?  In what ways to Mike and Kevin engage in interactive discourse?

05.34  Write a brief reflection on your own relational skills. Think here about all aspects of your work in schools – with your students, with your colleagues, with administration.  What do you feel that you do well? How do you know?  What do you feel that you would like to do better? How will you improve?  What do you want to learn more about?  Write a paragraph that summarizes factors that will move your leadership forward on the PRIME self-assessment rubric. Identify at least one specific descriptor that you selected. Where are you now for that descriptor? Where do you want to be?  Have your pre-observation conference completed by our next meeting on 18 November.  Create a system for collecting and storing your leadership project artifacts. Be prepared to share a bit about your system next time. ACTIVITY 5 HOMEWORK AND CLOSING REMARKS