NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
A Story of Ratios
Grade 8 – Module 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Session Objectives
• Examine the development of mathematical
understanding across the module using a focus on
concept development within the lessons.
• Identify the big ideas within each topic in order to
support instructional choices that achieve the lesson
objectives while maintaining rigor within the
curriculum.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development
Module Review
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Curriculum Overview of A Story of Ratios
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NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development
Module Review
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
L1: Why Move Things Around?
• We want to move things around in the plane to avoid
direct measurement. Are a pair of angles equal in
measure? Are the opposite sides of a rectangle the
same size?
• Students describe intuitively what motion would be
required to move a figure on the plane so that it rests
on top of a copy of the figure located somewhere else
on the plane. Students “map” a figure onto another.
Lesson 1, Concept Development
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Exploratory Challenge 1
Slide the original figure to the image (1) until they coincide. Slide the original
figure to (2), then flip it so they coincide. Slide the original figure to (3),
then turn it until they coincide.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 1, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L2: Definition of Translation and Three
Basic Properties
• Translation is defined as a transformation along a
vector.
• Translations have three basic properties:
• Translations map lines to lines, rays to rays, segments to
segments, and angles to angles.
• Translations preserve the lengths of segments.
• Translations preserve the measures of angles.
Lesson 2, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Exercise 2
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A Story of Ratios
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Exercise 2
G
F
D
A’
31˚
C’
B’
5
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E’
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 2, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
L3: Translating Lines
• What happens when a line is translated?
• The line and its image coincide.
OR
• The line and its image are parallel.
Lesson 3, Concept Development
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Exercises 1-4
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Exercises 1-4
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Exercises 1-4
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
• This summary is not accurate as it includes information about
corresponding angles. Which is a concept of Topic C. Please correct and
we will do the same.
Lesson 3, Student Debrief
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L4: Definition of Reflection and Basic
Properties
• A reflection is defined as a transformation that occurs
across a line.
• Reflections have three basic properties:
• Reflections map lines to lines, rays to rays, segments to
segments, and angles to angles.
• Reflections preserve the lengths of segments.
• Reflections preserve the measures of angles.
Lesson 4, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Example 4
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 4, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L5: Definition of Rotation and Basic
Properties
• A rotation is defined as a transformation that around
a center a given degree.
• Rotations have three basic properties:
• Rotations map lines to lines, rays to rays, segments to
segments, and angles to angles.
• Rotations preserve the lengths of segments.
• Rotations preserve the measures of angles.
Lesson 5, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Problem Set 1
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 5, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L6: Rotations of 180 Degrees
• Students know that a point (a, b) on the coordinate plane, after a rotation
of 180˚ will be located at (-a, -b).
• The point, center, and image of point are collinear.
Lesson 6, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Exit Ticket 1
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A Story of Ratios
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Exit Ticket 1
A’=(2, 4)
B’=(3, -1)
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
L7: Sequencing Translations
• Now that we know we can move things around in
the plane and they remain rigid, can we move things
back?
• Basis for conceptual understanding of congruence.
• The image of a figure can be mapped back onto its
original position by naming the vector that would
move it there. For example, a translation along
vector AB can be moved back to its original position
if you translate the image along vector BA.
Lesson 7, Concept Development
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Discussion
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
L8: Sequencing Reflections and
Translations
• A figure has been translated along a vector and
reflected across a line. Will its image be in the same
place if the figure had been reflected first and
translated second?
• The order in which we perform our basic rigid motions
has an effect on the final location of an image.
• Supports the development of conceptual
understanding of congruence.
Lesson 8, Concept Development
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Discussion
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Discussion
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L9: Sequencing Rotations
• What happens when a figure is rotated twice around the same center
compared to when it is rotated twice around two different centers?
Lesson 9, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Exploratory Challenge 2
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Exploratory Challenge 2
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 9, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
L10: Sequences of Rigid Motions
• Given two triangles in the plane, (one an image of the
other), can we describe a sequence of rigid motions
that would map one triangle onto the other?
• Focus on precision of language and descriptions of
sequences.
Lesson 10, Concept Development
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Exercise 4
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L11: Definition of Congruence and Some
Basic Properties
• Congruence is defined as a sequence of basic rigid
motions that maps one figure onto another.
• A congruence enjoys the same properties as the
individual basic rigid motions.
Lesson 11, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Exercise 1
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L12: Angles Associated with Parallel Lines
• When two lines are cut by a transversal, angles are
formed.
• Corresponding angles
• Alternate interior angles
• Alternate exterior angles
• When those lines are parallel, the angles formed are equal
in measure and the proof lies within our understanding of
rigid motions.
• Corresponding angles: translation
• Alternate interior and exterior angles: rotation
Lesson 12, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Exploratory Challenge 2
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Exploratory Challenge 2
The angles are equal in measure. A translation along the transversal will
map one angle onto the other.
The angles are equal in measure. A translation along the transversal will
map one angle onto the other.
The angles are equal in measure. A rotation around the midpoint of the
segment of the transversal between the parallel lines will map one angle
onto the other.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 12, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
L13: Angle Sum of a Triangle
• Students are shown 3 informal proofs as to why the
angle sum of a triangle is 180˚. (2 in this lesson and 1
in the next.)
• Students work through a proof of the angle sum of a
triangle that relies on understanding the basic rigid
motions and their properties.
Lesson 13, Concept Development
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Exploratory Challenge 2
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L14: More on the Angles of a Triangle
• How can we show that the measure of the exterior
angle of a triangle is equal to the sum of the measures
of the two remote interior angles?
• Use what we know about the sum of the angles of a
triangle.
• Use the fact that a straight angle is 180˚.
Lesson 14, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Exercise 4
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 14, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L15: Informal Proof of the Pythagorean
Theorem
• Square within a square proof.
• Applies knowledge learned in Module 1 with respect to
the first Law of Exponents.
• Students use what they know about congruence to verify
that four right triangles would have equal areas and apply
properties of congruence to make claims about angle
sums.
• Students know how to determine the measure of an
unknown angle of a straight angle.
Lesson 15, Concept Development
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Proof of Pythagorean Theorem
Looking at the outside
square only, the square with
side lengths (a+b), what is its
area?
Are the four triangles
congruent? How do you
know?
What is the area of one
triangle? What is the total
area of the four triangles?
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Proof of Pythagorean Theorem
Is the figure inside really a
square? How do we know?
What is the area of the
square in the figure?
How can we use areas to
show that the Pythagorean
Theorem is true?
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NYS COMMON CORE MATHEMATICS CURRICULUM
Proof of Pythagorean Theorem
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 15, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
L16: Applications of the Pythagorean
Theorem
• Traditional application of the Pythagorean theorem to determine the:
• length of the hypotenuse of a right triangle,
• distance between two points on the coordinate plane,
• finding height a ladder can reach as it leans again a wall,
• length of diagonal of a rectangle.
Lesson 16, Concept Development
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NYS COMMON CORE MATHEMATICS CURRICULUM
Exercise 3
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 16, Student Debrief
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development
Module Review
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Biggest Takeaway
Turn and Talk:
• What questions were answered for you?
• What new questions have surfaced?
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Key Points
• Experimental verifications of the basic rigid motions.
• Precision of language in descriptions of sequences.
• Immediate application of concept of congruence for
understanding angle relationships.
• Application of congruence and angle relationships to explain a
proof of the Pythagorean theorem.
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