07.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SCHOOL YEAR SESSION 7 2 DECEMBER 2015 CAVALIERI IS: A. A TYPE OF PASTA B. A MODEL OF CHEVY C. A GEOMETRIC PRINCIPLED. MIKE’S ITALIAN UNCLE

07.2 TODAY’S AGENDA  Volume and Cavalieri’s principle (Eureka Math TM G10 M3 L10)  “Big Ideas” discussion  Break  Modeling mentoring conversations: Mentoring and Coaching  Exploring the PRIME document  Homework and closing remarks

07.3 ACTIVITY 1 VOLUME OF PRISMS & CYLINDERS AND CAVALIERI’S PRINCIPLE EUREKA MATH TM GRADE 10 MODULE 3 LESSON 10 As you engage, consider: what aspects of Big Ideas 1 and 2 do you notice in the lesson? Record tangible examples of each to share at the lesson’s conclusion.

07.4 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Lesson goals  To understand the principle of parallel slices in the plane, and its 3- dimensional analogue, Cavalieri’s principle  To understand why the volume of a general cylinder is given by the formula (area of base) x height

07.5 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Remember from last week (Lesson 8):  The volume of a right cylinder is given by V = bh, where  b is the area of the base, and  h is the height (measured perpendicular to the base) Today, we are going to see why the same formula works for a general cylinder

07.6 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES  Complete the Opening Exercise in Lesson 10

07.7 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Take a look at Figure 1. Do you notice anything special about the three colored regions?

07.8 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Here is the same figure, with the cross-sectional lengths marked at a particular height h. What do you notice about those lengths?

07.9 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Principle of parallel slices in the plane:  If two planar figures of equal altitude have identical cross-sectional lengths at each height, then the figures have the same area. Turn and talk: why is this principle true?

07.10 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Principle of parallel slices in the plane:  If two planar figures of equal altitude have identical cross-sectional lengths at each height, then the figures have the same area. Turn and talk: what is the converse of the principle of parallel slices? Do you think the converse is true?

07.11 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES  Complete the Opening Exercise in Lesson 10

07.12 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Moving from 2-D to 3-D  What would be the three-dimensional analogue for the principle of parallel slices in the plane?

07.13 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Cavalieri’s principle:  Given two solids that are included between two parallel planes, if every plane parallel to the two planes intersects both solids in cross- sections of equal area, then the volumes of the two solids are equal.

07.14 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Quick write: (Think for 30 seconds; write for 90 seconds) Consider how Cavalieri’s principle is related to the principle of parallel slices. Explain each principle in your own words and how they are related.

07.15 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Turn and talk:  Assuming that two solids do meet the criteria of Cavalieri’s principle, does it imply that both solids are of the same type?  Does the principle only work if we compare two triangular prisms or two cylinders?  Does the principle apply if the two solids are different, e.g., one is a cylinder and one is a triangular prism, or if one solid is a right solid, while the other is an oblique solid?  Why or why not?

07.16 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Cavalieri’s principle:  Given two solids that are included between two parallel planes, if every plane parallel to the two planes intersects both solids in cross- sections of equal area, then the volumes of the two solids are equal.

07.17 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Cavalieri’s principle:

07.18 ACTIVITY 1 VOLUME AND CAVALIERI’S PRINCIPLE Turn and talk:  Explain why Cavalieri’s principle implies that the volume of any cylinder is equal to the area of its base times its height.  Hint. Consider a right cylinder and an oblique cylinder that have identical bases and the same height.

07.19 ACTIVITY 2 CROSS SECTIONS OF PYRAMIDS AND CONES Lesson goals  To understand the principle of parallel slices in the plane, and its 3- dimensional analogue, Cavalieri’s principle  To understand why the volume of a general cylinder is given by the formula (area of base) x height

07.20 ACTIVITY 2 “BIG IDEAS” DISCUSSION FROM NCTM ESSENTIAL UNDERSTANDINGS FOR GEOMETRY GRADES 9-12

07.21 ACTIVITY 2 “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.  Add some notes to the Big Ideas packet from last time regarding tonight’s lesson.

07.22 ACTIVITY 2 “BIG IDEAS” DISCUSSION  Big Idea 1. Working with diagrams is central to geometric thinking.  Essential Understanding 1a. A diagram is a sophisticated mathematical device for thinking and communicating.  Essential Understanding 1b. A diagram is a “built” geometric artifact, with both a history—a narrative of successive construction—and a purpose.  Big Idea 2. Geometry is about working with variance and invariance, despite appearing to be about theorems.  Essential Understanding 2a. Underlying any geometric theorem is an invariance— something that does not change while something else does.  Essential Understanding 2c. Examining the possible variations of an invariant situation can lead to new conjectures and theorems.

Break

07.24 ACTIVITY 3 MENTORING AND COACHING

07.25 ACTIVITY 3 MENTORING AND COACHING For this week’s work, you focused for the first time on real-time observation (not needing to participate as a learner). Discuss with your table:  What did real-time observation help you notice or wonder about that wasn’t as visible while you were acting also as a learner?  What aspects of the lesson were more difficult to see or connect to when you weren’t involved in the work?

07.26 ACTIVITY 3 MENTORING AND COACHING  The continuum from Mentoring to Coaching  Mentoring describes support and feedback that is more focused, localized, and individualized, with very specific feedback  Coaching is more likely to refer to longer-term change, a focus on systemic goals, and to happen across teachers  Coaching also frequently involves modeling lessons for and/or with teachers  Both Mentoring and Coaching are critical to the long-term success of a mathematics program and faculty

07.27 ACTIVITY 3 MENTORING AND COACHING The lesson you observed was designed to be a modeling experience for Mike’s preservice teachers (similar to a coaching experience).  Imagine a group of your colleagues were in the room watching the lesson.  Identify two aspects of the lesson that you would like to focus a coaching conversation around.  These could be things to reinforce, things to refine, or both.  How would you engage your colleagues in a discussion of these aspects of the lesson? Be specific – how would you open the discussion, and what would you want them to take away?

07.28 ACTIVITY 4 THE PRIME FRAMEWORK

07.29 ACTIVITY 4 THE PRIME FRAMEWORK Prepare a poster with the following:  A narrative summary of your project (a few sentences)  Your measures of success  What you have learned/accomplished so far  What questions, struggles, and obstacles do you anticipate

07.30 ACTIVITY 4 THE PRIME FRAMEWORK Gallery walk:  Briefly read each poster  Use your sticky notes to write a question or comment on each  Place your sticky note on the back of each poster

07.31 ADMINISTRIVIA Avoiding conflicts for the spring sessions… a proposal.  Current: Jan. 20, Feb. 17, Mar. 2, Mar. 16, Mar. 30, May 4  Proposed: Jan. 20, Feb. 10, Feb. 23, Mar. 9, Mar. 23, Apr. 20

07.32  What aspects of your leadership projects have the potential to engage other teachers?  Read pages 1-13 of It’s TIME. Consider the Supportive Conditions that are identified. Which of the bullet points identified on pages 16 and 17 would your leadership project have the potential to support, either directly or indirectly? ACTIVITY 5 HOMEWORK AND CLOSING REMARKS