Download Domain: Cluster: Level: Mathematical Content Standard: Featured

Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Understand similarity in terms of similarity transformations
High School: Geometry
Mathematical Content Standard:
1. Verify experimentally the properties of dilations given by a center and a
scale factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
Featured Mathematical
Practice:
Correlated WA Standard:
G.5.B
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
Standard Clarification/Example:
Given a center and a scale factor, verify experimentally, that when dilating a figure in a coordinate plane, a
segment of the pre-image that does not pass through the center of the dilation, is parallel to its image
when the dilation is performed. However, a segment that passes through the center remains unchanged.
Task Analysis:





Describe the effect of dilations on two-dimensional figures using
coordinates.
Construct a dilation of a line not passing through the center.
Construct a dilation of a line passing through the center.
Compare/Contrast dilations of a line not passing and passing
through the center.
Predict the outcome of a dilation on a line based on the
relationship to the center.
Vocabulary:
Prior







Dilation
Center of Dilation
Parallel
Pre-Image/Image
Segment
Ratio
Scale Factor
Explicit

Introductory

Pacing:
Sample Assessment Item:
Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Understand similarity in terms of similarity transformations
High School: Geometry
Mathematical Content Standard:
1. Verify experimentally the properties of dilations given by a center and a
scale factor:
b. The dilation of a line segment is longer or shorter in the ratio given by
the scale factor.
Featured Mathematical
Practice:
Correlated WA Standard:
G.5.B
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
Standard Clarification/Example:
Given a center and a scale factor, verify experimentally, that when performing dilations of a line segment, the
pre-image, the segment which becomes the image is longer or shorter based on the ratio given by the scale
factor.
Task Analysis:




Describe the effect of dilations on two-dimensional figures using
coordinates.
Construct dilations of a line segment given a center and various scale
factors.
Compare/Contrast dilations of a line segment given a center and
various scale factors.
Predict the lengths of line segments based on the ratio given by the
scale factor.
Vocabulary:
Prior

Prior Vocabulary
from G-SRT.1a
Explicit

Introductory

Pacing:
Sample Assessment Item:
Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Understand similarity in terms of similarity transformations
High School: Geometry
Mathematical Content Standard:
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
Featured Mathematical
Practice:
Correlated WA Standard:
G.3.B
MP.1 Make sense of problems and persevere in
solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique
the reasoning of others.
MP.8 Look for and express regularity in
repeated reasoning.
Standard Clarification/Example:


Use the idea of dilation transformations to develop the definition of similarity.
Given two figures determine whether they are similar and explain their similarity based on the
equality of corresponding angles and the proportionality of corresponding sides.
Task Analysis:



Pacing:
Understand that a two-dimensional figure is similar to another if the
second can be obtained from a sequence of rigid motions and a
dilation.
Construct various similarity transformations on two-dimensional figures
to develop the concept/definition of similarity.
Given two-dimensional figures with minimal information, deduce the
equality of corresponding angles and the proportionality of
corresponding sides to determine similarity.
Vocabulary:
Prior




Explicit
Vocabulary from GSRT.1b
Corresponding
Angles
Corresponding
Sides
Proportionality
Sample Assessment Item:

Definition of
Similarity
Introductory

Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Understand similarity in terms of similarity transformations
High School: Geometry
Mathematical Content Standard:
3. Use the properties of similarity transformations to establish the AA
criterion for two triangles to be similar.
Featured Mathematical
Practice:
Correlated WA Standard:
G.3.B
MP.1 Make sense of problems and persevere in
solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique
the reasoning of others.
MP.8 Look for and express regularity in
repeated reasoning.
Standard Clarification/Example:
Pacing:
Use the properties of similarity transformations to develop the criteria for proving similar triangles; AA .
Task Analysis:




Understand that a two-dimensional figure is similar to another if the
second can be obtained from a sequence of rigid motions and a
dilation.
Determine if two triangles are similar by using the properties of
similarity transformations.
Make observations to discover the minimal information required to
show that two triangles are similar.
Use the properties of similarity transformations to develop the criteria
for proving similar triangles; AA.
Vocabulary:
Prior

Vocabulary from GSRT.2
Explicit

Angle-Angle
Similarity (AA)
Introductory

Sample Assessment Item:
Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Prove theorems involving similarity
High School: Geometry
Mathematical Content Standard:
4. Prove theorems about triangles. Theorems include: a line parallel to
one side of a triangle divides the other two proportionally, and conversely;
the Pythagorean Theorem proved using triangle similarity.
Featured Mathematical
Practice:
Correlated WA Standard:
G.3.D
MP.1 Make sense of problems and persevere in
solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique
the reasoning of others.
MP.7 Look for and make use of structure.
Standard Clarification/Example:


Task Analysis:




Pacing:
Use AA, SAS, and SSS similarity theorems to prove triangles are similar.
Use triangle similarity to prove other theorems about triangles.
 Prove a line parallel to one side of a triangle divides the other two proportionally, and it’s
converse.
 Prove the Pythagorean Theorem using triangle similarity.
Determine and use AA, SAS, and SSS similarity theorems to prove
triangles are similar.
Use triangle similarity to prove Midsegment Theorem, Triangle
Proportionality Theorem and its converse.
Apply the Pythagorean Theorem to determine unknown side lengths in
right triangles.
Use two similar right triangles to prove the Pythagorean Theorem.
Vocabulary:
Prior





Midsegment
Proportionally
Transversal
Bisect
Pythagorean
Theorem
Explicit


Side-Side-Side
Similarity (SSS)
Side-Angle-Side
Sample Assessment Item:



Similarity (SAS)
Midsegment
Theorem
Triangle
Proportionality
Theorem
Converse of
Triangle
Proportionality
Theorem
Introductory

Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Prove theorems involving similarity
High School: Geometry
Mathematical Content Standard:
5. Use congruence and similarity criteria for triangles to solve problems
and to prove relationships in geometric figures.
Featured Mathematical
Practice:
Correlated WA Standard:
G.3.B
MP.1 Make sense of problems and persevere in
solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique
the reasoning of others.
MP.7 Look for and make use of structure.
Standard Clarification/Example:
Pacing:

Solve real world problems using congruence and similarity criteria for triangles.

Prove geometric figures, other than triangles, are similar and/or congruent.
Task Analysis:




Solve real world problems using congruence and similarity criteria for
triangles.
Show geometric figures are similar by using the definition of similarity.
Show geometric figures are congruent by using the definition of
congruency.
Prove geometric figures, other than triangles, are similar and/or
congruent.
Vocabulary:
Prior


Definition of
Congruency
Definition of
Similarity
Explicit

Introductory

Sample Assessment Item:
Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Define trigonometric ratios and solve problems involving right triangles
High School: Geometry
Mathematical Content Standard:
6. Understand that by similarity, side ratios in right triangles are
properties of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
Featured Mathematical
Practice:
Correlated WA Standard:
G.3.C
MP.3 Construct viable arguments and critique
the reasoning of others.
MP.4 Model with mathematics.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
Standard Clarification/Example:

Task Analysis:




Pacing:
Using a corresponding angle of similar right triangles, show that the relationships of the side ratios
are the same, which leads to the definition of trigonometric ratios for acute angles.
Apply the Pythagorean Theorem to determine unknown side lengths in
right triangles.
Explore the trigonometric ratios in a variety of similar and special right
triangles.
Determine the relationship of the side ratios for corresponding acute
angles are the same.
Define the trigonometric ratios for acute angles.
Vocabulary:
Prior






Acute Angle
Right Angle
Corresponding
Angle
Hypotenuse
Leg
Ratio
Explicit




Trigonometric
Ratios
Sine
Cosine
Tangent
Sample Assessment Item:





Opposite
Adjacent
Special Right
Triangles
(30 -60 -90 )
Triangle
(45 -45 -90 )
Triangle
Introductory

Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Define trigonometric ratios and solve problems involving right triangles
High School: Geometry
Mathematical Content Standard:
7. Explain and use the relationship between the sine and cosine of
complementary angles.
Featured Mathematical
Practice:
Correlated WA Standard:
G.3.E
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique
the reasoning of others.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in
repeated reasoning.
Standard Clarification/Example:
Explore the sine of an acute angle and the cosine of its complement and determine their relationship.
Task Analysis:




Use facts about complementary angles in a multi-step problem. (7th
grade math)
Identify the trigonometric ratios for the acute angles of a triangle.
Explore the relationship between the sine and cosine of
complementary angles.
Explain and use the relationship between the sine and cosine of
complementary angles.
Vocabulary:
Prior


Complementary
Angles
Prior Vocabulary
from G-SRT.6
Explicit

Introductory

Pacing:
Sample Assessment Item:
Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Define trigonometric ratios and solve problems involving right triangles
High School: Geometry
Mathematical Content Standard:
8. Use trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems.*
Featured Mathematical
Practice:
Correlated WA Standard:
MP.2 Reason abstractly and quantitatively.
G.3.E
G.3.D
MP.3 Construct viable arguments and critique
the reasoning of others.
MP. 4 Model with mathematics.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in
repeated reasoning.
Standard Clarification/Example:
Apply both trigonometric ratios and Pythagorean Theorem to solve application problems involving right
triangles.
Task Analysis:




Apply the Pythagorean Theorem to determine unknown side lengths in
right triangles.
Apply trigonometric ratios to determine unknown angle measures in
right triangles.
Apply Pythagorean Theorem and trigonometric ratios to solve for right
triangles.
Use trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems.
Vocabulary:
Prior

Vocabulary from GSRT.7
Explicit

Solve a Right
Triangle
Introductory

Pacing:
Sample Assessment Item:
Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Apply trigonometry to general triangles
High School: Geometry
Mathematical Content Standard:
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by
Featured Mathematical
Practice:
drawing an auxiliary line from a vertex perpendicular to the opposite side.
MP.1 Make sense of problems and persevere in
solving them.
Correlated WA Standard:
NONE
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique
the reasoning of others.
MP.4 Model with mathematics.
Standard Clarification/Example:
For a triangle that is not a right triangle, draw an auxiliary line from a vertex, perpendicular to the opposite side
and derive the formula, A=½ ab sin (C), for the area of a triangle, using the fact that the height of the triangle
is, h=a sin(C).
Task Analysis:



Vocabulary:
(7th
Solve mathematical problems involving area of a triangle.
grade
math)
Draw the auxiliary line from a vertex, perpendicular to the opposite side
and find the height in terms of the trigonometric ratio.
Apply the trigonometric ratios of the height to the formula for the area
of a triangle.
Prior






Height
Base
Area of Triangle
Perpendicular
Vertex
Vocabulary from GSRT.8
Explicit

Auxiliary Line
Pacing:
Sample Assessment Item:
Introductory

Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Apply trigonometry to general triangles
High School: Geometry
Mathematical Content Standard:
10. (+) Prove the Laws of Sines and Cosines and use them to solve
problems.
Featured Mathematical
Practice:
Correlated WA Standard:
NONE
MP1. Make sense of problems and persevere in
solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique
the reasoning of others.
MP.4 Model with mathematics.
Standard Clarification/Example:




Task Analysis:






Pacing:
Using trigonometry and the relationship among sides and angles of any triangle, such as h= a sin(C),
to prove the Law of Sines.
Using trigonometry and the relationship among sides and angles of any triangle and the Pythagorean
Theorem to prove the Law of Cosines.
Use the Laws of Sines to solve problems.
Use the Laws of Cosines to solve problems.
Vocabulary:
Use trigonometric ratios and the Pythagorean Theorem to solve right
Prior
triangles in applied problems.
 Vocabulary from GDraw the auxiliary line from a vertex, perpendicular to the opposite side
SRT.9
and find the height in terms of the trigonometric ratio.
Using trigonometry and the relationship among sides and angles of any Explicit
triangle, such as h= a sin(C), to prove the Law of Sines.
 Law of Sines
Using trigonometry and the relationship among sides and angles of any
 Law of Cosines
triangle and the Pythagorean Theorem to prove the Law of Cosines.
 Trigonometric
Use the Laws of Sines to solve problems.
Identity
Use the Laws of Cosines to solve problems.
Introductory
Sample Assessment Item:

(Theta)
Domain:
Cluster:
Level:
Similarity, Right Triangles, and Trigonometry G-SRT
Apply trigonometry to general triangles
High School: Geometry
Mathematical Content Standard:
11. (+) Understand and apply the Law of Sines and the Law of Cosines to
find unknown measurements in right and non-right triangles (e.g.,
surveying problems, resultant forces).
Featured Mathematical
Practice:
Correlated WA Standard:
NONE
MP.1 Make sense of problems and persevere in
solving them.
MP.4 Model with mathematics.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
Standard Clarification/Example:


Task Analysis:


Pacing:
Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in right
triangles.
Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in nonright triangles.
Understand and apply the Law of Sines and the Law of Cosines to find
unknown measures in right triangles.
Understand and apply the Law of Sines and the Law of Cosines to find
unknown measures in non-right triangles
Vocabulary:
Prior

Vocabulary from GSRT.10
Explicit

Introductory
Sample Assessment Item: