Download Geometry Unit 4

Geometry
Unit 4
Title
Suggested Time Frame
3rd
Similarity
4th
&
Six Weeks
Suggested Duration: 16 days
Guiding Questions
Big Ideas/Enduring Understandings
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 4
Updated November 12, 2015
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:
•
(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 4
Updated November 12, 2015
•
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
Knowledge and
Skills with Student
Expectations
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
systems,
District Specificity/ Examples
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).
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, onethird, 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
Geometry--Unit 4
Updated November 12, 2015
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
A&M Consolidated HS 03-04 Fall Unit 08 –
Similar Polygons
Example Reference:
Distance between points example
Partition a Segment Video
including finding
the midpoint.
Instead of using this to find only the midpoint: ((x1 + x2) / 2,
(y1 + y2) / 2)
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).
•
•
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G.3C*
Students should be able to identify the process and the
order of how the image was changed.
Teachers need to show examples on the coordinate plane
and off the plane.
Off the plane could be on patty paper and real world
examples. Off Coordinate Plane - Not limited to graphs
Geometry--Unit 4
Updated November 12, 2015
•
•
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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
(C) The student is
expected to
identify the
sequence of
transformations
that will carry a
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
How It Could Be Assessed:
Given pre image & image - students must list the
compositions applied
Misconception:
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)
Misconceptions:
• When naming figures order matters.
Geometry--Unit 4
Updated November 12, 2015
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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
•
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.
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.
G.7B
(B) The student is
expected to
apply the Angle- Big Idea - If triangles have two congruent angles, then they
are similar triangles therefore sides are proportional.
Angle criterion
to verify similar
Geometry--Unit 4
Updated November 12, 2015
**This is an extension of 7.A
Using the AA Theorem to Determine Similar
Triangles
triangles and
apply the
proportionality
of the
corresponding
sides to solve
problems.
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)
Mirror Mirror Project
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
Big Idea - prove theorems & apply to solve
Teachers should:
Geometry--Unit 4
Updated November 12, 2015
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•
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altitude
Angle-Angle
Similarity
Theorem
Angle-SideAngle
theorems by utilizing
a variety of methods
such as coordinate,
transformational,
axiomatic and
formats such as twocolumn, paragraph,
flow chart.
(A) The student is
expected to
prove theorems
about similar
triangles,
including the
Triangle
Proportionality
theorem, and
apply these
theorems to
solve problems;
and
(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.
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
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•
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Students need to:
• Complete proofs of AA Similarity Postulate, SSS
Similarity Theorem, SAS Similarity Theorem, and
the Triangle Proportionality Theorem.
• Find missing sides and angles
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)
Geometry--Unit 4
Updated November 12, 2015
Similarity
Theorem
geometric
mean
hypotenuse
right triangle
Side-Side-Side
Similarity
Theorem
similar
triangles
theorem
Triangle
Proportionality
theorem
Mirror Mirror Project
IXL – Triangle Proportionality Theorem
Stating and Proving the Triangle
Proportionality Theorem
Similarity in Right Triangles - Kuta
Similarity in Right Triangles - Anderson
Proofs Involving Similar Triangles
Examples
Geometry--Unit 4
Updated November 12, 2015
Give the appropriate theorem or postulate that can be
used to prove the triangles similar and write the
similarity statement.
G.8B
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.
Geometry--Unit 4
Updated November 12, 2015
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
• Draw the altitude
• Identify similar triangles created
Geometry--Unit 4
Updated November 12, 2015
Examples:
Geometry--Unit 4
Updated November 12, 2015