presentation: module focus geometry module 3

NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Functions

A Story of FunctionsGrade 10 Geometry Module 3

Extending to Three Dimensions

N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Functions

Participant Poll• Classroom teacher• Math trainer or coach• Principal or school leader• District representative / leader• Other

Session Objectives• Participants will understand the development of

geometric reasoning in two dimensions and how that development is extended in support of like reasoning in three dimensions.

• Participants will enrich their knowledge and experience in order to implement Module 3 with confidence and success.

AgendaToday• Topic A - Area• Topic B – Volume Part I• Lunch• Topic B – Volume Part II

Tomorrow• Topic B – Volume Part III• End of Module Assessment• Discussion for Implementation

Module 3: Extending to Three DimensionsModule Overview• 2 Topics• 13 Lessons• 15 days• No mid-module assessment• Geometric Measurement and Dimension

• G-GMD.A.1, 2+, 3, and 4• Modeling with Geometry

• G-MG.A.1, 2, and 3

Module 3: Extending to Three DimensionsTopic A

• Tie up the loose ends surrounding knowledge about area.• Follows the same progression seen in earlier grades.• Begin using informal limit arguments.

Topic B• Three-Dimensional Space• Figures in three-dimensions• Making sense of three-dimensional figures by looking at 2-dimensional figures

within• Cavalieri’s Principle• Cool application

Topic A: AreaTopic Overview• Lessons 1-4• Mature our understanding of area.• Finding the area of a curved figure.• Use of informal limit arguments.• Formally examine the properties of area.• Scaling and its impacts on area.• Prove the area of a disk.

Lesson 1: What Is Area?What is the area of the rectangle below whose side lengths measure units by units?

Lesson 1: What Is Area?Exploratory ChallengeWhat is the area of the rectangle shown whose side lengths measure units by units? Use the unit squares on the graph to guide your approximation. Explain how you determined your answer.

Lesson 1: What Is Area?YOUR TURN!Finding the area of a rectangle whose side lengths are irrational.• 1’s: Complete a table of approximations of the side lengths

and area of rectangles that are less than those of the given triangle. (A table is provided for you on P.3 of the Participant Materials Packet.)

• 2’s: Complete a table of approximations of the side lengths and area of rectangles that are greater than those of the given rectangle. (A table is provided for you on P.4 of the Participant Materials Packet.)

Lesson 1: What Is Area?Approximating the Percent Error of our approximated areas.• We know that the true area is between and .

• What is the maximum absolute error?• Using the above approximations, what is the maximum percent error?

• Find the maximum percent error for your next three area approximations.

Lesson 1: What Is Area?If it takes one can of paint to cover a square unit in the coordinate plane, how many cans of paint are needed to paint the region within the curved figure?

Participant Materials

Lesson 1: What Is Area?YOUR TURN!Take a few moments to complete the Exit Ticket on P.5 of your Participant Materials. Be prepared to share your solution with the group.

Lesson 2: Properties of AreaYOUR TURN!Complete Exploratory Challenge 1-4 on P. 6-8 of your Participant Materials packet, then we will derive and record the properties of area.

1. Area is…2. A polygonal region is the union of all…3. Congruent regions…4. The area of the union of two regions is…5. The area of the difference of two regions…

Lesson 2: Properties of AreaYOUR TURN!Apply the properties of area in completing Problem Set #3 on P. 9 of the Participant Materials packet. Be prepared to share your solution.

Lesson 3: The Scaling Principle for AreaBased on your knowledge about dilations, form a conjecture about the relationship between the area of an original figure and its scaled image with respect to the scale factor used.

Does your conjecture hold true for all plane figures?

Polygon Polygon

Lesson 3: The Scaling Principle for AreaThe Scaling Principle for TrianglesIf similar triangles and are related by a scale factor of , then the respective areas are related by a factor of .

Triangle

Lesson 3: The Scaling Principle for AreaThe Scaling Principle for PolygonsIf similar polygons and are related by a scale factor of , then their respective areas are related by a factor of .

Polygon Polygon

Lesson 3: The Scaling Principle for AreaThe Scaling Principle for AreaIf similar figures and are related by a scale factor of , then their respective areas are related by a scale factor of .

Lesson 3: The Scaling Principle for Area

Lesson 3: The Scaling Principle for AreaYOUR TURN!Take a few moments to complete Problem Set #6 and #8 on P. 10 in your Participant Materials packet and be prepared to share your solution and explain.

Lesson 4: Proving the Area of a DiskOpening ExerciseThe following image is of a regular hexagon inscribed in circle with radius . Find a formula for the area of the hexagon in terms of the length of a side, and the distance from the center to a side.

Participant Materials packet P. 11

Lesson 4: Proving the Area of a DiskExample 1 – Inscribed Regular Polygons

Participant Materials P. 11-12

Lesson 4: Proving the Area of a DiskExample 1 – Circumscribed Regular Polygons

Participant Materials P. 13

Lesson 4: Proving the Area of a DiskDiscussionHow will the area of compare to the area of the disk?

How will the area of compare to the area of the disk?

How does the area of the circle compare to and ? Why?

What happens as the value of gets larger and larger (as it approaches )?

Lesson 4: Proving the Area of a DiskLimit (description). Given an infinite sequence of numbers, , to say that the

limit of the sequence is means, roughly speaking, that when the index is very large, then is very close to . This is often denoted as, “As , .”

Area of a circle (description). The area of a circle is the limit of the areas of the inscribed regular polygons as the number of sides of the polygons approaches infinity.

Lesson 4: Proving the Area of a DiskYOUR TURN!Take a few moments to complete the Exit Ticket for Lesson 4 on P. 15 in your Participant Materials packet and be prepared to share your solution and explain.

Summing Up Topic ALet’s ReviewWhat major concepts and/or themes did you notice in Topic A? How do you think those topics and themes will apply as we move into three-dimensions?

Topic B: VolumeTopic Overview• 9 lessons (5-13)• Basic properties of 3-dimensional space• Categorizing three-dimensional solids• Examining cross-sections of solids• Properties of volume• Developing volume formulas• The impacts of scaling on volume• Cavalieri’s Principle• Geometric modeling

Lesson 5: Three-Dimensional SpaceThe properties of Points, Lines and Planes in 3-Dimensional Space are provided on P. 16-18 of the Participant Materials packet. Examine the properties then share some visual modelling ideas that can be used to scaffold these properties for the students in your classroom.

Lesson 5: Three-Dimensional SpaceYOUR TURN!Discuss and complete Problem Set #1 provided on P. 18 of the Participant Materials.

Lesson 6: General Prisms and Cylinders and Their Cross-SectionsYOUR TURN!Given the Exploratory Challenge on P. 19 of your Participant Materials, use each description to sketch the figure described.

Slice or Cross-section?

Lesson 6: General Prisms and Cylinders and Their Cross SectionsEXTENSIONWhat can we claim about all cross-sections of a given general cylinder?Consider a triangular prism.

Lesson 7: General Pyramids and Cones and Their Cross-SectionsThe General ConeLet be a region in a plane , and be a point not in . The cone with base and vertex is the union of all segments , for all points in .

Lesson 7: General Pyramids and Cones and Their Cross-SectionsExample 1 (Alternative Extension):

Lesson 7: General Pyramids and Cones and Their Cross-Sections

Lesson 7: General Pyramids and Cones and Their Cross-SectionsExample 2:

In the following triangular pyramid, a plane passes through the pyramid so that it is parallel to the base and results in the cross section . The altitude from is drawn; the intersection of the altitude with the base is and the intersection of the altitude with the cross section is . If the distance from to is , the distance from to is , and the area of is , what is the area of ?

Participant Materials P.20

Lesson 7: General Pyramids and Cones and Their Cross-SectionsGeneral Cone Cross-section Theorem• If two general cones have the same base area and the same height, then

the cross-sections for the general cones the same distance from the vertex have the same area.

Lesson 7: General Pyramids and Cones and Their Cross-SectionsYOUR TURN!Take a few moments to complete the Exit Ticket on P. 22 of your Participant Materials packet and be prepared to share your solution.

Lesson 8: Definition and Properties of VolumeArea Properties Volume Properties

1. The area of a set in 2-dimensions is a number greater than or equal to zero that measures the size of the set and not the shape.

1. The volume of a set in 3-dimensions …

2. The area of a rectangle is given by the formula . The area of a triangle is given by the formula . A polygonal region is the union of finitely many non-overlapping triangular regions and has area equal to the sum of the areas of the triangles.

2. A right rectangular prism has volume given by the formula . A right prism is the union of…

…is a number greater than or equal to zero that measures the size

of the set and not the shape

…finitely many non-overlapping right rectangular or right

triangular prisms and has volume equal to the sum of the volumes

of those prisms.

3. Congruent regions have the same area.

3. Congruent solids…

4. The area of the union of two regions is the sum of the areas minus the area of the intersection:

4. The volume of the union of two solids is …

…have the same volume.

the sum of the volumes minus the volume of the

intersection.

5. The area of the difference of two regions where one is contained in the other is the difference of the areas:If , then

5. The volume of the difference of two solids where one is contained in the other is… the difference of the volumes: If , then

6. The area of a region can be estimated by using polygonal regions and so that is contained in and is contained in . Then .

6. The volume of a solid can be estimated … by using right prism solids and so that .

Then .

Lesson 8: Definition and Properties of Volume

YOUR TURN!Take a few moments to complete the Exit Ticket on P. 23 of your Participant Materials packet and be prepared to share and explain your solution.

Lesson 9: Scaling Principle for VolumesWhat does it mean for two solids in three-dimensional space to be similar?

How do you think the volumes of similar solids are related?

Test your conjecture by completing Exercise 4 on P. 24-25 of your Participant Materials packet.

Lesson 9: Scaling Principle for VolumesThinking back to Lesson 3 (Scaling Principle for Area), how do you think the volume of a scaled figure is related to the volume of the original if it is scaled in three perpendicular directions with scale factors and ?

Take a few moments to complete Exercise 6 on P. 25-26 of your Participant Materials. Be prepared to share your answers.

Lesson 9: Scaling Principle for VolumesYOUR TURN!Take a few moments to complete the Exit Ticket for Lesson 9 on P. 27 of your Participant Materials packet. Compare your answers with a neighbor and be prepared to share with the group.

Lesson 10: The Volume of Prisms and Cylinders and Cavalieri’s PrincipleWhat are some observations and/or conjectures regarding the red, green, and blue planar regions shown below?

Principle of Parallel Slices: If two planar figures of equal altitude have identical cross-sectional lengths at each height, then the

regions of the figures have the same area.

Lesson 10: The Volume of Prisms and Cylinders and Cavalieri’s PrincipleCavalieri’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.

Lesson 10: The Volume of Prisms and Cylinders and Cavalieri’s Principle

Lesson 10: The Volume of Prisms and Cylinders and Cavalieri’s PrincipleRecall the Lesson 6 Extension where we showed that all the cross-sections of a general cylinder are congruent. What does this conclusion help us to further conclude? (Use the picture below as a hint.)

Lesson 10: The Volume of Prisms and Cylinders and Cavalieri’s PrincipleYOUR TURN!Take a few moments to complete Problem Set #1 on P. 28 of your Participant Materials packet. Be prepared to share your answer.

Lesson 11: The Volume Formula of a Pyramid and ConeDiscuss with a neighbor how this explains the in the volume formula. Be prepared to share with the group!

Lesson 11: The Volume Formula of a Pyramid and ConeHow do we compute the volume of the general cone shown below with a base area and height ?

Lesson 11: The Volume Formula of a Pyramid and ConeYOUR TURN!Take a few moments to complete Exercise 4 on P. 29 in your Participant Materials packet. Be prepared to share your solution.

Lesson 11: The Volume Formula of a Pyramid and ConeYOUR TURN!Take a few moments to complete Problem Set #12 on P. 30 in your Participant Materials packet. Be prepared to share your solution.

Lesson 12: The Volume Formula of a SphereSPHERE: Given a point in the -dimensional space and a number , the sphere with center and radius is the set of all points in space that are distance from the point .

SOLID SPHERE OR BALL: Given a point in the -dimensional space and a number , the solid sphere (or ball) with center and radius is the set of all points in space whose distance from the point is less than or equal to .

Lesson 12: The Volume Formula of a SphereDiscussionSolid hemisphere has radius . Right circular cone has height and a circular base with radius , and its vertex lies in the plane with the bases of the hemisphere and cylinder. Cylinder has a circular base with radius and its height is also .

Lesson 12: The Volume Formula of a SphereThe Solid HemisphereFind the radius of the cross-sectional disk of the solid hemisphere, , in terms of and .

Lesson 12: The Volume Formula of a SphereThe Right Circular ConeFind the radius of the cross-sectional disk of the right circular cone, , in terms of and .

Lesson 12: The Volume Formula of a SphereThe Right Circular Cylinder

Lesson 12: The Volume Formula of a SphereReviewing the facts:• The solids have the same height, .• At any given height, , the sum of the areas of the cross-sectional disks of

the hemisphere and the cone are equal to the area of the cross-section of the cylinder.

Lesson 12: The Volume Formula of a SphereExtension: The Formula for Surface Area of a Sphere

Lesson 12: The Volume Formula of a SphereYOUR TURN!Take a few moments to complete Problem Set #9 and #10 on P. 33 of your Participant Materials packet. Be prepared to share your solution and strategies.

Lesson 13: How do 3D Printers Work?Exercise 1Sketch five evenly spaced, horizontal cross-sections made with the following figure.

Participant Materials P. 34 http://commons.wikimedia.org/wiki/File%3ATorus_illustration.png; By Oleg Alexandrov (self-made, with MATLAB) [Public domain], via Wikimedia Commons. Attribution not legally required.

Diameter inches

Height inch

Design an approximation of the ring above, with the given

dimensions, made up of five equally spaced layers.

Lesson 13: How do 3D Printers Work?Five Equally Spaced Layers

Lesson 13: How do 3D Printers Work?Example 1Sketch the cross-sections of the coffee mug shown at the heights indicated in the image.

To watch a 3D printer in action, watch the video below.

https://www.youtube.com/watch?v=29yHrWrs1ok

Lesson 13: How do 3D Printers Work?

Lesson 13: How do 3D Printers Work?For more on the capabilities of 3D Printing, watch the video below.

http://computer.howstuffworks.com/3-d-printing.htm

Lesson 13: How do 3D Printers Work?YOUR TURN!Take a few moments to complete the Exit Ticket for Lesson 13 on P. 36 of your Participant Materials packet. Be prepared to share your solution.

Summing up Topic BLet’s ReviewWhat major concepts and/or themes did you notice in Topic B?

End-of-Module Assessment• Do the problems in the End-of-Module

Assessment• As you work, think about the following:

• Which lesson(s) does this assessment item tie to?• Is there vocabulary that students may struggle

with?• Can this item be used as part of a quiz for Topic A?

Key Themes of Module 3: Extending to Three Dimensions • Built upon students knowledge of congruence and similarity from Modules

1 and 2.• Matures students’ understandings of area and volume to be numbers that

represent the size of a set in the plane or in space without giving reference to the shape of the set.

• Three-dimensional solids are examined as they are made up of two-dimensional parts (i.e. cross-sections).

• Measure irregular figures by squeezing between approximations of known figures.

• Measure irregular figures by comparing to known figures (Cavalieri’s Principle).

Biggest Takeaway• What is your biggest takeaway with respect to Module

• How can you support successful implementation at your school/s given your role?

presentation: module focus geometry module 3

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