NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
Grade 10 Geometry Module 3
Extending to Three Dimensions
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Participant Poll
•
•
•
•
•
Classroom teacher
Math trainer or coach
Principal or school leader
District representative / leader
Other
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Today
• 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Module 3: Extending to Three Dimensions
Module 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Module 3: Extending to Three Dimensions
Topic 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Topic A: Area
Topic 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: What Is Area?
What is the area of the rectangle below whose side lengths measure 3 units by
5 units?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: What Is Area?
5
3
What is the area of the rectangle below whose side lengths measure units by
3
4
units?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 1: What Is Area?
Exploratory Challenge
What is the area of the rectangle
shown whose side lengths measure
3 units by 2 units? Use the unit
squares on the graph to guide your
approximation. Explain how you
determined your answer.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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.)
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: What Is Area?
Approximating the Percent Error of our approximated areas.
• We know that the true area is between 2.38 and 2.7.
• 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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
P. 4
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Properties of Area
YOUR 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.
2.
3.
4.
5.
Area is…
A polygonal region is the union of all…
Congruent regions…
The area of the union of two regions is…
The area of the difference of two regions…
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Properties of Area
YOUR 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: The Scaling Principle for Area
Based 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 𝑃
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Polygon 𝑄
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: The Scaling Principle for Area
The Scaling Principle for Triangles
If similar triangles 𝑆 and 𝑇 are related by a scale factor of 𝑟, then the
respective areas are related by a factor of 𝑟 2 .
Triangle 𝑆
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Triangle 𝑇
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: The Scaling Principle for Area
The Scaling Principle for Polygons
If similar polygons 𝑃 and 𝑄 are related by a scale factor of 𝑟, then their
respective areas are related by a factor of 𝑟 2 .
Polygon 𝑃
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Polygon 𝑄
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: The Scaling Principle for Area
The Scaling Principle for Area
If similar figures 𝐴 and 𝐵 are related by a scale factor of 𝑟, then their
respective areas are related by a scale factor of 𝑟 2 .
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: The Scaling Principle for Area
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: The Scaling Principle for Area
YOUR 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: Proving the Area of a Disk
Opening Exercise
The following image is of a regular hexagon inscribed in circle 𝐶 with radius 𝑟.
Find a formula for the area of the hexagon in terms of the length of a side, 𝑠,
and the distance from the center to a side.
Participant Materials packet P. 11
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: Proving the Area of a Disk
Example 1 – Inscribed Regular Polygons
Participant Materials P. 11-12
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: Proving the Area of a Disk
Example 1 – Circumscribed Regular Polygons
Participant Materials P. 13
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: Proving the Area of a Disk
Discussion
How will the area of 𝑃𝑛 compare to the area of the disk?
How will the area of 𝑃𝑛′ compare to the area of the disk?
How does the area of the circle compare to 𝐴𝑟𝑒𝑎 𝑃𝑛 and 𝐴𝑟𝑒𝑎(𝑃𝑛′ )? Why?
What happens as the value of 𝑛 gets larger and larger (as it approaches ∞)?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: Proving the Area of a Disk
Limit (description). Given an infinite sequence of numbers, 𝑎1 , 𝑎2 , 𝑎3 , … , to
say that the limit of the sequence is 𝐴 means, roughly speaking, that when
the index 𝑛 is very large, then 𝑎𝑛 is very close to 𝐴. This is often denoted
as, “As 𝑛 → ∞, 𝑎𝑛 → 𝐴.”
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: Proving the Area of a Disk
YOUR 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Summing Up Topic A
Let’s Review
What 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 threedimensions?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Topic B: Volume
Topic 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5: Three-Dimensional Space
The 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5: Three-Dimensional Space
YOUR TURN!
Discuss and complete Problem Set #1 provided on P. 18 of the Participant
Materials.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: General Prisms and Cylinders and Their
Cross-Sections
YOUR TURN!
Given the Exploratory Challenge on P. 19 of your Participant Materials, use
each description to sketch the figure described.
Slice or Cross-section?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: General Prisms and Cylinders and Their
Cross Sections
EXTENSION
What can we claim about all cross-sections of a given general cylinder?
Consider a triangular prism.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7: General Pyramids and Cones and Their
Cross-Sections
The General Cone
Let 𝐵 be a region in a plane 𝐸, and 𝑉 be a point not in 𝐸. The cone with base
𝐵 and vertex 𝑉 is the union of all segments 𝑉𝑃, for all points 𝑃 in 𝐵.
V
V
B
E
© 2012 Common Core, Inc. All rights reserved. commoncore.org
B
E
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7: General Pyramids and Cones and Their
Cross-Sections
Example 1 (Alternative Extension):
V
B'
E'
B
E
Participant Materials P. 19
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: General Pyramids and Cones and Their
Cross-Sections
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: General Pyramids and Cones and Their
Cross-Sections
Example 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 △ 𝐴′𝐵′𝐶′. The
altitude from 𝑉 is drawn; the intersection of the
altitude with the base is 𝑋, and the intersection of
the altitude with the cross section is 𝑋′. If the
distance from 𝑋 to 𝑉 is 18 𝑚𝑚, the distance from
𝑋′ to 𝑉 is 12 𝑚𝑚, and the area of △ 𝐴′𝐵′𝐶′ is
28 𝑚𝑚2 , what is the area of △ 𝐴𝐵𝐶?
Participant Materials P.20
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7: General Pyramids and Cones and Their
Cross-Sections
General 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.
B'
B
© 2012 Common Core, Inc. All rights reserved. commoncore.org
C'
C
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: General Pyramids and Cones and Their
Cross-Sections
YOUR 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Definition and Properties of Volume
Area 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…
…finitely many non-overlapping right
rectangular or right triangular prisms
and has volume equal to the sum of
the volumes of those prisms.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
…is a number greater than or equal to
zero that measures the size of the set
and not the shape
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Definition and Properties of Volume
Area Properties
3. Congruent regions have the same
area.
Volume Properties
3. Congruent solids…
…have the same volume.
4. The area of the union of two regions is 4. The volume of the union of two solids
the sum of the areas minus the area
is … the sum of the volumes minus
the volume of the intersection.
of the intersection:
𝑽𝒐𝒍 𝑨 ∪ 𝑩 = 𝑽𝒐𝒍 𝑨 + 𝑽𝒐𝒍 𝑩 − 𝑽𝒐𝒍 𝑨 ∩ 𝑩
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Definition and Properties of Volume
Area Properties
5. The area of the difference of two
regions where one is contained in the
other is the difference of the areas:
If , then
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Volume Properties
5. The volume of the difference of two
solids where one is contained in the
other is… the difference of the
volumes: If 𝐴 ⊆ 𝐵, then
𝑉𝑜𝑙 𝐵 − 𝐴 = 𝑉𝑜𝑙 𝐵 − 𝑉𝑜𝑙(𝐴)
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Definition and Properties of Volume
Area Properties
Volume Properties
6. The area of a region can be
estimated by using polygonal regions
and so that is contained in and is
contained in .
Then .
2
–5
5
–2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
6. The volume of a solid can be
estimated … by using right prism
solids 𝑆 and 𝑇 so that 𝑆 ⊆ 𝑊 ⊆ 𝑇.
Then 𝑉𝑜𝑙(𝑆) ≤ 𝑣 ≤ 𝑉𝑜𝑙(𝑇).
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Definition and Properties of Volume
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Definition and Properties of Volume
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9: Scaling Principle for Volumes
What 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9: Scaling Principle for Volumes
Thinking 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 𝑟, 𝑠, and 𝑡?
Take a few moments to complete Exercise 6 on P. 25-26 of your Participant
Materials. Be prepared to share your answers.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9: Scaling Principle for Volumes
YOUR 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: The Volume of Prisms and Cylinders and
Cavalieri’s Principle
What 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.
Participant Materials P. 28
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: The Volume of Prisms and Cylinders 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 crosssections of equal area, then the volumes of the two solids are equal.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: The Volume of Prisms and Cylinders and
Cavalieri’s Principle
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: The Volume of Prisms and Cylinders and
Cavalieri’s Principle
Recall 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.)
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: The Volume of Prisms and Cylinders and
Cavalieri’s Principle
YOUR TURN!
Take a few moments to complete Problem Set #1 on P. 28 of your Participant
Materials packet. Be prepared to share your answer.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11: The Volume Formula of a Pyramid and
Cone
1
Discuss with a neighbor how this explains the in the volume formula. Be
3
prepared to share with the group!
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11: The Volume Formula of a Pyramid and
Cone
How do we compute the volume of the general cone shown below with a base
area 𝐴 and height ℎ?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11: The Volume Formula of a Pyramid and
Cone
YOUR TURN!
Take a few moments to complete Exercise 4 on P. 29 in your Participant
Materials packet. Be prepared to share your solution.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11: The Volume Formula of a Pyramid and
Cone
YOUR TURN!
Take a few moments to complete Problem Set #12 on P. 30 in your Participant
Materials packet. Be prepared to share your solution.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The Volume Formula of a Sphere
SPHERE: Given a point 𝐶 in the 3-dimensional space and a number 𝑟 > 0, the
sphere with center 𝐶 and radius 𝑟 is the set of all points in space that are
distance 𝑟 from the point 𝐶.
SOLID SPHERE OR BALL: Given a point 𝐶 in the 3-dimensional space and a number
𝑟 > 0, the solid sphere (or ball) with center 𝐶 and radius 𝑟 is the set of all
points in space whose distance from the point 𝐶 is less than or equal to 𝑟.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The Volume Formula of a Sphere
Discussion
Solid hemisphere 𝐻 has radius 𝑅. Right circular cone 𝑇 has height 𝑅 and a
circular base with radius 𝑅, and its vertex lies in the plane with the bases of
the hemisphere and cylinder. Cylinder 𝑆 has a circular base with radius 𝑅 and
its height is also 𝑅.
Participant Materials P. 31
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The Volume Formula of a Sphere
The Solid Hemisphere
Find the radius of the cross-sectional disk of the solid hemisphere, 𝑟1 , in terms
of 𝑅 and ℎ.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The Volume Formula of a Sphere
The Right Circular Cone
Find the radius of the cross-sectional disk of the right circular cone, 𝑟2 , in
terms of 𝑅 and ℎ.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The Volume Formula of a Sphere
The Right Circular Cylinder
𝐴𝑟𝑒𝑎 𝐷1 = 𝜋𝑅2 − 𝜋ℎ2
𝐴𝑟𝑒𝑎 𝐷2 = 𝜋ℎ2
𝐴𝑟𝑒𝑎 𝐷3 = 𝜋𝑅2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The Volume Formula of a Sphere
Reviewing the facts:
• The solids have the same height, 𝑅.
• At any given height, ℎ, 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The Volume Formula of a Sphere
Extension: The Formula for Surface Area of a Sphere
A1
A11 A A
A8 A 6
A2 A A
3
A4 A5
9
10
7
Participant Materials P. 32
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The Volume Formula of a Sphere
YOUR 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13: How do 3D Printers Work?
Exercise 1
Sketch five evenly spaced, horizontal cross-sections made with the following
figure.
Diameter 5 inches
Height 1 inch
Design an approximation of the ring
above, with the given dimensions,
made up of five equally spaced
layers.
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 13: How do 3D Printers Work?
Five Equally Spaced Layers
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 13: How do 3D Printers Work?
Example 1
Sketch the cross-sections of the coffee mug shown at the heights indicated in
the image.
Participant Materials P. 35
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 13: How do 3D Printers Work?
To watch a 3D printer in action, watch the video below.
https://www.youtube.com/watch?v=29yHrWrs1ok
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Summing up Topic B
Let’s Review
What major concepts and/or themes did you notice in Topic B?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
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?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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 twodimensional 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).
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Biggest Takeaway
•
What is your biggest takeaway with respect to Module 3?
•
How can you support successful implementation at your
school/s given your role?
© 2012 Common Core, Inc. All rights reserved. commoncore.org