Download Geometry Curriculum

Geometry Curriculum
Paterson Charter School for Science and Technology
2014-2015
Unit 2 Summary: Similarity, Right Triangles and Trigonometry
Essential Questions
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Big Ideas
What are the criteria for triangles to be similar?
How does this relate to transformations?
How can a line drawn parallel to one of
the sides of a triangle be used to solve
problems involving triangles? How can
the median be used?
How can the side of a triangle be partitioned
into segments of a given ratio? How can this
information be used to solve problems
involving similar triangles?
How does the perpendicular bisector of a
segment relate to isosceles triangles? How can
constructions be used to verify this?
What are the properties of isosceles triangles
and how can they be used to solve problems?
What are the criteria for triangles to be
congruent? How does this relate to
transformations?
How does congruency relate to similarity?
How can similarity and congruence be used to
solve problems and/or prove statements about
or properties of triangles?
Geometry Curriculum
Unit 2: Similarity, Right Triangles and Trigonometry
Unit 2 Summary: Similarity, Right Triangles and Trigonometry
Learning Outcomes (CCSS)
G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
G.SRT.1a 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.
G.SRT.1bThe dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.SRT.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.
G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity
G.SRT.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.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles
G.SRT.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.
G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
Apply trigonometry to general triangles
G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side.
G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
G.SRT.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).
Geometry Curriculum
Unit 2 Similarity, Right Triangles and Trigonometry
Unit 2 Summary: Similarity, Right Triangles and Trigonometry
25.
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29.
Local Standards (LS) #24 are taught in this unit.
Verify experimentally the properties of dilations given by a center and a scale factor.
Know that 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.
Know the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
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 a l l corresponding pairs of sides.
30.
31.
32
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35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
Identify and use properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems about triangles: a line parallel to one side of a triangle divides the other two proportionally and its converse.
Prove the Pythagorean Theorem using triangles similarity
Recognize the characteristics of congruent figures recognize corresponding figures and their corresponding parts.
Prove two triangles congruent using the SSS, SAS, ASA Postulate and AAS Theorem.
Use triangle congruence and corresponding parts of congruent triangles to prove that parts of two triangles are congruent.
Prove geometric figures, other than triangles, are similar and/or congruent.
Use properties of isosceles triangles to show that right triangles are congruent and prove right triangles are congruent using the Hypotenuse-Leg Theorem
Identify congruent overlapping triangles and prove two triangles congruent using other congruent triangles.
Identify characteristics of midsegments and use properties of midsegments in order to solve problems.
Identify and use properties of perpendicular bisectors, medians and altitudes to solve problems.
Find the sum of the measures of the interior and exterior angles of a polygon
Use relationships among sides and angles and diagonals of parallelograms in order to solve problems
Determine properties of similar polygons in order to identify and apply them in problem solving.
Apply use of AA Similarity Postulate, SSS Similarity Theorem and SAS Similarity Theorem to prove triangles similar.
Use triangle similarity to find indirect measurements.
Find and use relationships in similar right triangles in order to solve problems.
Use geometric software to explore the trigonometric ratios of sine, cosine, and tangent.
Use the sine, Cosine and Tangent ratio to find missing side lengths and angle measures in a right triangle.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems .
Geometry Curriculum
Unit 2: Similarity, Right Triangles and Trigonometry
Unit 2 Summary: Similarity, Right Triangles and Trigonometry
Prior Knowledge Expected
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Draw, construct, and describe geometrical figures and describe the relationships between them.
Understand congruence and similarity using physical models, transparencies, or geometry software.
Assessment Summary
1.
(Scope and Sequence on the next page)
Geometry Curriculum
Unit 2 Similarity, Right Triangles and Trigonometry
Unit 1 Scope and Sequence: Number System Fluency
Periods*
CCSS
LS
Student Friendly Language
Resources
Textbook
Links
1
G.SRT.
1
S25

Verify experimentally the properties of dilations given by a
center and a scale factor.
Other
Pearson Concept Byte 9-6
Illustrative Mathematics:
Dilating a Line
Engage NY Module 2:
Lessons 1-11
Assessmen
Skill Check 25
t
1
G.SRT.
1a
S26
Know that 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.
Textbook
Pearson Concept Byte 9-6
Links
Illustrative Mathematics:
Dilating a Line
Other
Engage NY Module 2:
Lessons 1-11
Assessmen
Skill Check 26
t
1
G.SRT.
1b
Geometry Curriculum
S27
Know the dilation of a line segment is longer or shorter in the
ratio given by the scale factor.
Textbook
Pearson Concept Byte 9-6
Links
Illustrative Mathematics:
Dilating a Line
Other
Engage NY Module 2:
Lessons 1-11
Unit 2: Similarity, Right Triangles and Trigonometry
Assessmen
Skill Check 27
t
Textbook
Links
1
G.SRT.2
S28
Use the idea of dilation transformations to develop the definition
of similarity.
Pearson 9-7
 Inside Mathematics
Performance Task:
Hopewell Geometry
Rhombuses
 Illustrative
Mathematics: Are
they Similar?
Other
Engage NY Module 2:
Lessons 12-24
Assessmen
Skill Check 28
t
Textbook
1
G.SRT.2
S29
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.
Links
Pearson 9-7
 Inside Mathematics
Performance Task:
Hopewell Geometry
Rhombuses
 Illustrative
Mathematics: Are
they Similar?
Geometry Curriculum
Unit 2 Similarity, Right Triangles and Trigonometry
1
1
G.SRT.3
G.SRT.4
Geometry Curriculum
S30
S31
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 29
Textbook
Pearson 9-7
Links
IIustrative Mathematics
Similar triangles
Identify and use properties of similarity transformations to establish
the AA criterion for two triangles to be similar.
Other
Engage NY Module 2:
Lesson 12-24
Assessment
Skill Check 30
Textbook
Pearson 7-5
Links
 Illustrative
Mathematics:
Joining two midpoints of
sides of a triangle
Pythagorean Theorem
Tangent Line to Two
Circles
Bank Shot
Extensions, Bisections
and Dissections in a
Rectangle Folding a
square into thirds
Other
Engage NY Module 2:
Lesson 1-11
Assessment
Skill Check 31
Prove theorems about triangles: a line parallel to one side of a
triangle divides the other two proportionally and its converse.
Unit 2: Similarity, Right Triangles and Trigonometry
1
G.SRT.4
Textbook
Pearson 8-1
Links
 Illustrative
Mathematics:
Joining two midpoints of
sides of a triangle
Pythagorean Theorem
Tangent Line to Two
Circles
Bank Shot
Extensions, Bisections
and Dissections in a
Rectangle Folding a
square into thirds
Other
Engage NY Module 2:
Lesson 1-11
Assessment
Skill Check 32
S32 Prove the Pythagorean theorem using triangle similarity.
Textbook
Pearson
Links
1
1
G.SRT.5
G.SRT.5
Geometry Curriculum
S33
S34
Recognize the characteristics of congruent figures recognize
corresponding figures and their corresponding parts
Prove two triangles congruent using the SSS, SAS, ASA Postulate
and AAS Theorem.
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 33
Textbook
Pearson 4-2, 4-3
Links
Unit 2 Similarity, Right Triangles and Trigonometry
1
1
G.SRT.5
G.SRT.5
35
36
Use triangle congruence and corresponding parts of congruent
triangles to prove that parts of two triangles are congruent.
Use properties of isosceles triangles to show that right triangles are
congruent and prove right triangles are congruent using the
Hypotenuse-Leg Theorem
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 34
Textbook
Pearson 4-4
Links
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 35
Textbook
Pearson 4-5, 4-6
Links
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 36
Textbook
Links
1
1
G.SRT.5
G.SRT.5
Geometry Curriculum
37
38
Prove geometric figures, other than triangles, are similar and/or
congruent.
Identify congruent overlapping triangles and prove two triangles
congruent using other congruent triangles.
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 37
Textbook
Pearson 4-7
Links
Unit 2: Similarity, Right Triangles and Trigonometry
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 38
Textbook
Links
1
G.SRT.5
39
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 39
Textbook
Links
1
G.SRT.5
40
Identify characteristics of midsegments and use properties of
midsegments in order to solve problems.
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 40
Textbook
1
G.SRT.5
41
Links
Identify and use properties of perpendicular bisectors, medians and
altitudes to solve problems.
Other
Assessment
1
G.SRT.5
Geometry Curriculum
42
Engage NY Module 2:
Lessons 12-24
Skill Check 41
Find the sum of the measures of the interior and exterior angles of a Textbook
polygon
Links
Unit 2 Similarity, Right Triangles and Trigonometry
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 42
Textbook
Links
1
G.SRT.5
43
Use relationships among sides and angles and diagonals of
parallelograms in order to solve problems
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Textbook
1
G.SRT.5
44
Determine properties of similar polygons in order to identify and
apply them in problem solving.
Links
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Textbook
1
G.SRT.5
45
Links
Apply use of AA Similarity Postulate, SSS Similarity Theorem and
SAS Similarity Theorem to prove triangles similar.
Other
Assessment
1
Engage NY Module 2:
Lessons 12-24
Skill Check 45
Textbook
G.SRT.5
46
Use triangle similarity to find indirect measurements.
Links
Geometry Curriculum
Unit 2: Similarity, Right Triangles and Trigonometry
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 46
Textbook
1
G.SRT.6
47
Links
Find and use relationships in similar right triangles in order to solve
problems.
Other
Assessment
Engage NY Module 2:
Lessons 12-24
Skill Check 47
Textbook
Links
1
2
2
G.SRT.6
G.SRT.7
G.SRT.8
48
49
50
Use geometric software to explore the trigonometric ratios of sine,
cosine, and tangent
Other
Engage NY Module 2:
Lessons 12-24
Assessment
Skill Check 48
Textbook
Pearson 8-3
Links
Use the Sine, Cosine and Tangent ratio to find missing side lengths
and angle measures in a right triangle.
Other
Use trigonometric ratios and the Pythagorean Theorem to solve
right triangles in applied problems.
Assessment
Skill Check 49
Textbook
Pearson
8-1, 8-2, 8-3, 8-4 CB 8-4
Links
Other
Geometry Curriculum
Engage NY Module 2:
Lessons 12-24
Skill Check 50
Unit 2 Similarity, Right Triangles and Trigonometry
Assessment
Geometry Curriculum
Engage NY Module 2:
Lessons 12-24
Unit 2: Similarity, Right Triangles and Trigonometry