Download Geometry 2016

Geometry
2016-17 ~ Unit 5
Title
Suggested Time Frame
3rd & 4th Six Weeks
Suggested Duration: 16 days
Similarity
CISD Safety Net Standards: NONE in this unit
Big Ideas/Enduring Understandings
Guiding Questions
Module 11
• Similarity and Transformation can be used to solve real-world
problems
Module 11
• How does a dilation transform a figure?
• How can similarity transformations be used to find out how two figures
are similar?
• If you know two figures are similar, what can you determine about
measures of corresponding angles and lengths?
• How can you show that two triangles are similar?
Module 12
• Similar Triangles can be used to solve real-world problems.
Module 12
• When a line parallel to one side of a triangle intersects the other two
sides, how does it divide those sides?
• How do you find the point on a directed line segment that partitions
the given segment in a given ratio?
• How can you use similar triangles to solve problems?
• How does the altitude to the hypotenuse of a right triangle help you
use similar right triangles to solve problems?
Vertical Alignment Expectations
TEA Vertical Alignment Chart Grades 5-8, Geometry
Sample Assessment Question
Coming Soon...
Geometry ~ Unit 5
Updated October 17, 2016
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Geometry
2016-17 ~ Unit 5
The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper
depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the
suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the
district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. Portions of the District Specificity and
Examples are a product of the AAMS found within the Region XI Mathematics Support pages.
Ongoing TEKS
Math Processing Skills
G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding.
The student is expected to:
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(A) apply mathematics to problems arising in everyday life, society, and the
workplace;
(B) use a problem-solving model that incorporates analyzing given
information, formulating a plan or strategy, determining a solution, justifying
the solution, and evaluating the problem-solving process and the
reasonableness of the solution;
•
(C) select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math,
estimation, and number sense as appropriate, to solve problems;
Focus is on application
Students should assess which tool to apply rather than trying only one or all
(D) communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams, graphs, and language
as appropriate;
(E) create and use representations to organize, record, and communicate
mathematical ideas;
•
(F) analyze mathematical relationships to connect and communicate
mathematical ideas; and
•
Students are expected to form conjectures based on patterns or sets of examples
and non-examples
(G) display, explain, and justify mathematical ideas and arguments using
precise mathematical language in written or oral communication
•
Precise mathematical language is expected.
Geometry ~ Unit 5
Updated October 17, 2016
•
Students should evaluate the effectiveness of representations to ensure they are
communicating mathematical ideas clearly
Students are expected to use appropriate mathematical vocabulary and phrasing
when communicating ideas
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Geometry
2016-17 ~ Unit 5
Knowledge and
Skills with Student
Expectations
District Specificity/ Examples
G.2 Coordinate and
Transformational
Geometry. The
student uses the
process skills to
understand the
connections between
algebra and
geometry and uses
the one- and twodimensional
coordinate systems
to verify geometric
conjectures.
(A) The student is
expected to
determine the
coordinates of a
point that is a
given fractional
distance less
than one from
one end of a line
segment to the
other in oneand twodimensional
coordinate
G.2A*
A big idea for this student expectation (SE) is that there are an infinite amount of
points on a line between two endpoints and students will need to find any point that
is a fractional distance away.
One-dimensional:
For any fractional distance, students need to multiply the difference of one
coordinate by the fractional distance (k).
Vocabulary
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coordinate
distance
line segment
midpoint
point
endpoint
Suggested Resources
Resources listed and
categorized to indicate
suggested uses. Any
additional resources must
be aligned with the TEKS.
HMH Geometry
Unit 4
Example Reference:
Distance between
points example
Partition a Segment
Video
Students will need to use this formula to find any fractional distance: (x2 - x1) *k + x1
Examples:
On a number line, point A is located at -3 and point B is located at 21. Find points that
are located one-fourth, one-third, three-fifths, etc. of the distance between the
endpoints.
Two-dimensional:
For any other fractional distance, students need to multiply the difference of each set
of coordinates by the fractional distance (k) and then add that distance to the original
coordinates (x1, y1).
Students will need to use this new formula to find any fractional distance: ((x2 - x1) *k
+ x1 RUN, (y2 - y1) * k + y1) RISE
Instead of using this to find only the midpoint: ((x1 + x2) / 2, (y1 + y2) / 2)
Geometry ~ Unit 5
Updated October 17, 2016
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systems,
including finding
the midpoint.
Geometry
2016-17 ~ Unit 5
Examples:
Point P has coordinates (-8, 5) and Point Q has coordinates (4, -1). Find points that are
located one-fourth, one-third, three-fifths, etc. of the distance between the
endpoints.
Teachers should initially make a connection between the midpoint formula and
general form which can apply to any ratio.
Example Reference:
Construct a partition of a segment (see video in resources column to the right)
G.3 Coordinate and
Transformational
Geometry. The
student uses the
process skills to
generate and
describe rigid
transformations
(translation,
reflection, and
rotation) and nonrigid transformations
(dilations that
preserve similarity
and reductions and
enlargements that do
not preserve
similarity).
(C) The student is
expected to
identify the
sequence of
transformations
that will carry a
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G.3C*
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Teachers need to show examples on the coordinate plane and off the plane.
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Off the plane could be on patty paper and real world examples. Off Coordinate Plane Students should be able to identify the process and the order of how the image was
changed.
dilation
enlargement
(scale factor
>1)
image
point of
rotation
pre-image
reduction
(0<scale
factor<1)
reflection
rotation
scale factor
similarity
transformation
translations
Web Resources
Dilations and Similarity
Example
Dilations
Activity/Worksheet
Dilations and Scale
Factor Worksheet
Not limited to graphs
How It Could Be Assessed:
Given pre image & image - students must list the compositions applied
Misconception:
Geometry ~ Unit 5
Updated October 17, 2016
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given pre-image
onto an image
on and off the
coordinate
plane; and
G.7 Similarity, Proof,
and Trigonometry.
The student uses the
process skills in
applying similarity to
solve problems.
(A) The student is
expected to
apply the
definition of
similarity in
terms of a
dilation to
identify similar
figures and their
proportional
sides and the
congruent
corresponding
angles; and
Geometry
2016-17 ~ Unit 5
Students assume that performing the transformations in any order will result in same
result
G.7A
Big Idea - identify similar figures, their proportional sides and congruent angles
Prior Knowledge - refer back to dilations covered through transformation (w/ any
point as center of dilation), similar figures
Teachers should show:
• Characteristics of similar figures
• Using proportions to find missing sides (Cross products are equal)
• Identifying corresponding angles
Students should:
• Know key attributes similar figures (congruent angles & corresponding sides
are proportional)
• Find missing angles and side lengths
• Know how to apply a scale factor ( > 1 and <1)
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Angle-Angle
Criterion
congruent
congruent
corresponding
angles
corresponding
angles
corresponding
sides
dilation
proportional
proportionality
similar figures
similar
triangles
similarity
Web Resources
Similarity, Congruency,
and Transformations
Activity Set
Khan Academy
Resource – Similarity
Postulates
Proofs with Similar
Triangles Examples
Similarity Activities
and Lessons from
NSA.gov
Misconceptions:
• When naming figures order matters.
• Corresponding angles should be in the same order in a figure.
Examples: Use properties of triangles and the similarity statement to find the
measures of the numbered angles.
Geometry ~ Unit 5
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Geometry
2016-17 ~ Unit 5
Using the given measurements, find the scale factor (similarity ratio) of each of the
following sets of similar figures, comparing smaller to larger; then comparing larger to
smaller.
(B) The student is
expected to
apply the AngleAngle criterion
to verify similar
triangles and
apply the
proportionality
of the
corresponding
G.7B
Big Idea - If triangles have two congruent angles, then they are similar triangles
therefore sides are proportional.
Teacher should:
• Show example of similar triangles with AA
• Show examples where congruent angles might be from vertical angles,
parallels cut by a transversal (prior knowledge)
Geometry ~ Unit 5
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sides to solve
problems.
Geometry
2016-17 ~ Unit 5
Students should:
• Identify corresponding angles
• Understand the AA theorem
• Similar triangles - proportional sides
• Set up proportions and solve for missing sides
• Find the 3rd angle for both triangles
Examples:
Misconceptions The sides in SSS and SAS must be proportional now (and not
congruent like they were with proving triangles congruent)
G.8 Similarity, Proof, G.8A
and Trigonometry.
The student uses the
process skills with
deductive reasoning
to prove and apply
theorems by utilizing
a variety of methods
such as coordinate,
transformational,
axiomatic and
formats such as twocolumn, paragraph,
flow chart.
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Big Idea - prove theorems & apply to solve
Teachers should:
Complete proofs of AA Similarity Postulate, SSS Similarity Theorem, SAS
Similarity Theorem, and the Triangle Proportionality Theorem.
Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides of the
triangle, then the line divides these two sides proportionally
Geometry ~ Unit 5
Updated October 17, 2016
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altitude
Angle-Angle
Similarity
Theorem
Angle-SideAngle
Similarity
Theorem
geometric
mean
hypotenuse
right triangle
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(A) The student is
expected to prove
theorems about
similar triangles,
including the
Triangle
Proportionality
theorem, and apply
these theorems to
solve problems; and
•
Students need to:
• Complete proofs of
Postulate, SSS
SAS Similarity
Triangle
Theorem.
• Find missing sides
AA Similarity
Similarity Theorem,
Theorem, and the
Proportionality
and angles
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Geometry
2016-17 ~ Unit 5
Side-Side-Side
Similarity
Theorem
similar
triangles
theorem
Triangle
Proportionality
theorem
Misconceptions
• Must have the included angle to be true for SAS
• The sides in SSS and SAS must be proportional now (and not congruent
like they were with proving triangles congruent)
Examples
Geometry ~ Unit 5
Updated October 17, 2016
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Geometry
2016-17 ~ Unit 5
Give the appropriate theorem or postulate that can be used to prove the
triangles similar and write the similarity statement.
Geometry ~ Unit 5
Updated October 17, 2016
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(B) The student is
expected to identify
and apply the
relationships that
exist when an
altitude is drawn to
the hypotenuse of a
right triangle,
including the
geometric mean, to
solve problems.
G.8B
Geometry
2016-17 ~ Unit 5
Big Idea - identify and apply relationships of an altitude drawn to the
hypotenuse to solve problems
Teachers need to Show:
Geometric Mean - It is the nth root of the product of n numbers. That means
you multiply the numbers together, and then take the nth root, where n is the
number of values you just multiplied.
Altitude on a Right Triangle- The measure of the altitude drawn from the
vertex of the right angle of a right triangle to its hypotenuse is the geometric
mean between the measures of the two segments of the
hypotenuse. In terms of our triangle, this theorem simply states what we have
already shown:
since AD is the altitude drawn from the right angle of our right triangle to its
hypotenuse, and CD and DB are the two segments of the hypotenuse.
Students Need To:
• Be able to calculate geometric mean
• Work backwards from geo mean to a side
Geometry ~ Unit 5
Updated October 17, 2016
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Geometry
2016-17 ~ Unit 5
Draw the altitude
Identify similar triangles created
Examples:
Geometry ~ Unit 5
Updated October 17, 2016
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