NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
A Story of Geometry
Grade 8 to Grade 10 Coherence
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Objectives
• Articulate and model the instructional approaches to
teaching the content.
• Examine the coherence of topics and lessons from
grade 8 to grade 10.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Participant Poll
•
•
•
•
•
Classroom teacher
School leader
Principal
District leader
BOCES representative
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
• Congruence and Rigid Motions
• Grade 8: Basic Rigid Motions
•
Translation, Reflection, Rotation
• Grade 10: Basic Rigid Motions
•
Translation, Reflection, Rotation
• Congruence
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Transformations in Geometry under the CCSS
• Transformations, specifically rigid motions, serve as the foundation of the
concept of congruence
• Why is congruence defined in terms of rigid motions?
• To avoid having to directly measure objects:
• Are the opposite sides of a rectangle really equal in length?
• Are two angles positioned differently in space really of equal measure?
• To develop an intuitive sense of congruence,
leading to a definition that can be used with all
figures in the plane-not just triangles and polygons.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Same Size
&
Same Shape
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 8
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Translation
• Translation is defined as a motion that “slides” figures along a vector.
• A vector is a segment in the plane with a designated starting point and endpoint.
AB
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BA
NYS COMMON CORE MATHEMATICS CURRICULUM
Activity
• Draw the following on a piece of paper:
• A line,
• A ray,
• A segment,
• A point,
• An angle,
• A curved figure,
• A simple drawing of your choice.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Properties of Translation
• We have experimentally verified that a translation:
• Maps lines to lines, rays to rays, segments to segments, and angles to angles.
• Preserves lengths of segments.
• Preserves angles measures of angles.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Translation of Lines
• Some properties of translation are highlighted.
• Example: What properties can we discuss about translated lines?
• There are two possible scenarios:
1) A line and its translated image coincide (when the vector belongs to the line or is
parallel to the line):
1) A line and its translated image will be parallel (when the vector is not parallel to the
line):
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
A Sequence of Translations
• Imagine life without an “undo” button on your smart device or computer!
• We want to make sure that when we move things around in the plane, we can put
them back where they belong, or “undo” the motion.
• For that reason, we show students how a translation along a vector
undone by translating along a vector BA.
AB can be
• This is the beginning of the concept of congruence. It shows that a sequence of
two translations can map a figure onto itself.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Activity
• Take out your paper and transparency.
• This time, reflect each of the images you drew by “flipping” your
transparency across the line you drew.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Properties of Reflection
• We have experimentally verified that a reflection:
• Maps lines to lines, rays to rays, segments to segments, and angles to angles.
• Preserves lengths of segments.
• Preserves angles measures of angles.
• Additional property that is verified:
• When you connect a point and it’s reflected image, the segment is perpendicular
to the line of reflection.
• Not only is the line of reflection perpendicular to the segment, but it bisects the segment.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Activity
• Take out your paper and transparency.
• This time, rotate each of the images you drew by placing your finger on top
of the point you drew and carefully rotate your transparency in one
direction and then the other.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Properties of Rotation
• We have experimentally verified that a rotation:
• Maps lines to lines, rays to rays, segments to segments, and angles to angles.
• Preserves lengths of segments.
• Preserves angles measures of angles.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Congruence
• With each new rigid motion that is learned, students immediately begin
sequencing the motion with a known motion.
• For example:
• The first sequence is two translations.
• Once reflection is learned, students sequence two reflections. Then, students sequence a
translation and a reflection.
• Once rotation is learned, students sequence two rotations. Then, students sequence a
translation and a rotation, or a rotation and a reflection, etc.
• Congruence is defined in terms of a sequence of rigid motions, performed
using a transparency, that shows the mapping of one figure onto another.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Rigid Motions in Grade 10
• Students enter Grade 10 with an intuitive sense of congruence and have
experimentally verified properties of rigid motions
•
•
They know that “same size, same shape” is not a precise way of describing congruence
Defining congruence with the use of rigid motions captures all types of figures
• In Grade 10, students formalize the concepts from Grade 8 through
language
•
•
•
The visual/experiential understanding of how each rigid motion actually “works” is put into
explicit parameters
Students think about the plane and the rigid motions in the plane more abstractly
Constructions are used in the application of rigid motions
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Grade 10: Reflection
• In Grade 8, students have used transparencies to experimentally verify the
properties of a reflection AND that the line of reflection is the
perpendicular bisector of any segment that joins a pair of corresponding
points between the figure and its image
• In Grade 10, students clearly define reflection and how to:
i.
ii.
Determine the line of reflection by construction
Reflect a figure across a line by construction.
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Determining the Line of Reflection
• Use the construction of a perpendicular bisector to determine the line of
reflection for the following figures:
A
E
C
B
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G
F
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Determining the Line of Reflection
Y
A
C
E
G
Z
B
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F
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Grade 10: Mapping over the Line of Reflection
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Grade 10: Mapping over the Line of Reflection
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Grade 10: Rotation
• In Grade 8, students experimented with a model of a rotation, spinning
figures on transparencies to verify that rotations were indeed distance
preserving and angle preserving.
• In Grade 10, students clearly define rotation and learn to:
i. Determine the center of rotation
ii. Determine the angle of rotation
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Determining the Angle of Rotation
A
C
B
C'
A'
B'
D
Y
D'
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Determining the Angle of Rotation
A
C
B
C'
A'
B'
D
Y
D'
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Grade 10: Determining the Center of Rotation
B
A
B'
A'
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Grade 10: Determining the Center of Rotation
A
B
B'
A'
© 2012 Common Core, Inc. All rights reserved. commoncore.org
!
GQCO.2
.
!A!transformation!! !of!the!plane!is!a!function!that!
assigns!to!each!point!!
!of!the!plane!a!unique!point!!
(! )!in!the!plane.!!! A Story of Ratios
NYS COMMON
CORE MATHEMATICS
CURRICULUM
!
The!
counterclockwise,half9plane,of,a,ray,CP,is!the!halfQplane!of!line!! " !that!lies!to!
the!left!as!you!move!along!! " !in!the!direction!from!! !to!! .!!!
!
GQCO.4
.
!For!0 < ! < 180,!the!rotation,of,! ,degrees,around,the,center!
! !is!the!transformation!! ! ,! !of!the!plane!defined!as!follows:!
!
1. For!the!center!point!! ,!! ! ,! ! = ! ,!and!
2. For!any!other!point!! ,!! ! ,! ! !is!the!point!! !that!lies!in!the!
counterclockwise!halfQplane!of!ray!! " !such!that!! " = ! " !and!
∠! " #! = !! ˚.!!
!
A!rotation,of,0˚,around,the,center,! !is!the!identity!transformation,!i.e.,!for!all!
points!! !in!the!plane,!! ! ,! (! ) = ! .!!
!
A!rotation,of,180˚,around,the,center,! !is!the!composition!of!two!rotations!by!
90˚!about!the!center!! .!!It!is!also!the!transformation!that!maps!a!point!! !(other!
than!! )!to!the!other!endpoint!of!the!diameter!of!circle!with!center!! !and!radius!
! " .!!
!
For!! > 180,!a!rotation,of,! ˚,around,the,center,! !is!any!composition!of!three!
or!more!rotation!such!that!each!rotation!is!less!than!or!equal!to!a!90˚!rotation!
and!whose!angle!measures!sum!to!! ˚.!!For!example,!a!rotation!of!240˚!is!equal!
to!the!composition!of!three!rotations!by!80˚!about!the!same!center,!or!the!
composition!of!three!rotations!by!100˚,!100˚,!and!40˚!about!the!same!center,!
© 2012 Common Core, Inc. All rights reserved. commoncore.org
or!the!composition!of!240!rotations!by!1!degree!about!the!same!center.!!!
Grade 10: Rotation
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Grade 10: Translation
• In Grade 8, students experimented with a model of a translations, sliding
figures on transparencies to verify that translations were distance
preserving and angle preserving.
• In Grade 10, students clearly define translation and learn to:
i. Apply a translation by constructing parallel lines
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Applying a Translation
• Given the experience students enter Grade 10 with, they can visualize the
image of the figure under a translation, provided the vector.
P1
P1
P3
P2
A
B
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P2
P3
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Applying a Translation
• To apply the translation, we must construct the line parallel to each side in
the direction and at a distance equal to the length of the vector.
P1
P2
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P3
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Applying a Translation
Follow the instructions to construct the line parallel to AB through P.
P
A
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B
l
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Applying a Translation
Line PQ is parallel to line AB.
C1
P
A
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Q
l
B
C2
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Translating a Segment
• The translation of a segment might look like this:
P1
Q1
P1
A
P2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A
B
P2
Q2
B
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Translating a Triangle
• The translation of a triangle might look like this:
P1
P1
Q1
P3
P2
A
B
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P3
P2
A
Q2
Q3
B
NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 10: Translation
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Congruence
• Once students are comfortable with rigid motions, they study the link
between the concept of rigid motions and congruence
Congruent. Two figures in the plane are congruent if
there exists a finite composition of basic rigid motions that
maps one figure onto the other figure.
• We want students to be able to use the language around congruence in a
clear way
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Congruence
Sample Question:
Why can’t a triangle be congruent to a
quadrilateral?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Congruence
Sample Question:
Why can’t a triangle be congruent to a
quadrilateral?
Sample Answer:
A triangle cannot be congruent to a
quadrilateral because there is no rigid
motion that takes a figure with three
vertices to a figure with four vertices.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Coherence
• It is important that students leave Grade 8 with a solid, intuitive
understanding of the rigid motions
• The physical manipulation of and visual understanding of rigid motions in Grade 8
needs be put into careful language in Grade 10
• Properties of rigid motions that make obvious sense need are married with
construction, and eventually used in reasoning
• The “careful use of language” is mentioned frequently in Grade 10.
• Ultimately, we want students to understand that Geometry exists as a axiomatic
system- that the establishment of a new fact comes strictly from basic
assumptions or existing facts
• These assumptions and existing facts appear throughout Module 1, and certainly
in the topic of Rigid Motions.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Biggest Takeaway
• A solid understanding of how rigid motions behave in Grade 8 will lay the
groundwork for Grade 10.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Key Points
• The hands-on, experiential understanding and experimental verification of
properties in Grade 8 are formalized through language and construction in
Grade 10.
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