Download Geometry Mathematics Curriculum Guide

Geometry Mathematics Curriculum Guide
2015 – 2016
Unit 6: Trigonometry and Special Right Triangles
Time Frame: 14 Days
Primary Focus
This topic extends the idea of triangle similarity to indirect measurements. Students develop properties of special right triangles, and use properties of similar
triangles to develop their understanding of trigonometric ratios. These ideas are then applied to find unknown lengths and angle measurements.
Common Core State Standards for Mathematical Practice
Standards for Mathematical Practice
How It Applies to this Topic…
MP1 - Make sense of problems and persevere in solving them.
Analyze given information to develop possible strategies for solving the
problem.
Use observations and prior knowledge (stated assumptions, definitions, and
previous established results) to make conjectures and construct arguments.
Use a variety of methods to model, represent, and solve real-world problems.
Generalize the process to create a shortcut which may lead to developing rules
or creating a formula.
MP3 - Construct viable arguments and critique the reasoning of
others.
MP4 - Model with mathematics.
MP8 - Look for and express regularity in repeated reasoning.
Unit 6
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
2015 – 2016
Stage 1 Desired Results
Transfer Goals
Students will be able to independently use their learning to…
Use the properties of special right triangles to solve real-world geometric situations.
Solve geometric problems involving the basic trigonometric ratios of sine, cosine, and tangent.
UNDERSTANDINGS
Students will understand that…
The angles in right triangles are related to the ratios of the side lengths.
The sine and cosine of complementary angles are related.
Right triangles properties can be applied to solve problems.
Meaning Goals
ESSENTIAL QUESTIONS
How do the ratios of the side lengths of right triangles relate to the angles in the
triangle?
What is the relationship of the cosine and the sine of two complementary
angles?
What does it mean to "solve" a triangle?
Acquisition Goals
Students will know and will be skilled at…
Naming the sides of right triangles as related to an acute angle
Recognizing that if two right triangles have a pair of acute, congruent angles that the triangles are similar
Comparing common ratios for similar right triangles and develop a relationship between the ratio and the acute angle leading to the trigonometry ratios
Using the relationship between the sine and cosine of complementary angles
Identifying the sine and cosine of acute angles in right triangles
Identifying the tangent of acute angles on right triangles
Explaining how the sine and cosine of complementary angles are related to each other
Recognizing which methods could be used to solve right triangles in applied problems
Solving for an unknown angle or side of a right triangle using sine, cosine, and tangent
Applying right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
Unit 6
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
2015 – 2016
Stage 1 Established Goals: Common Core State Standards for Mathematics
Cluster: Standard(s)
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.
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.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
★Specific modeling standard (versus an example of the modeling Standard for Mathematical Practice).
2008 Geometry Standard connection: G.3.C,D & E
Explanations, Examples, and Comments
Generalize this theorem to prove that the figure formed by joining consecutive midpoints of sides of an arbitrary
quadrilateral is a parallelogram. (This result is known as the Midpoint Quadrilateral Theorem or Varignon’s
Theorem.)
Use cardboard cutouts to illustrate that the altitude to the hypotenuse divides a right triangle into two triangles
that are similar to the original triangle. Then use AA to prove this theorem. Then, use this result to establish the
Pythagorean relationship among the sides of a right triangle and thus obtain an algebraic proof of the
Pythagorean Theorem.
Prove that the altitude to the hypotenuse of a right triangle is the geometric mean of the two segments into
which its foot divides the hypotenuse.
Prove the converse of the Pythagorean Theorem, using the theorem itself as one step in the proof. Some
students might engage in an exploration of Pythagorean Triples (e.g., 3-4-5, 5-12-13, etc.), which provides an
algebraic extension and an opportunity to explore patterns.
What students should know prior to this unit and may need to be reviewed
Fluency with ratios and proportional reasoning
Fluency with dilations
Unit 6
Clover Park School District 2015-2016
Stage 3
MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine
which of the materials below best meet student
instructional needs. All materials listed may not be
needed.
Holt Geometry Lesson 5-7 The Pythagorean Theorem
Holt Geometry Lesson 5-8 Applying Special Right
Triangles
Holt Geometry Lesson 8-1 Similarity in Right
Triangles
Holt Geometry Lesson 8-2 Trigonometric Ratios
Holt Geometry Lesson 8-3 Solving Right Triangles
Holt Geometry Lesson 8-4 Angles of Elevation and
Depression
Or use EngageNY Lessons Listed Below
EngageNY Geometry Module 2: Lesson 21,24-30
Supplemental Resources
Pythagorean Theorem
Discovering Geometry 9.1The Theorem of Pythagoras
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Geometry Mathematics Curriculum Guide
2015 – 2016
Recognize a situation’s connection to a mathematical model
Basic ability to mathematically support a prediction or hypothesis
Discovering Geometry 9.2 The Converse of the
Pythagorean Theorem
Explanations, Examples, and Comments
Students may use applets to explore the range of values of the trigonometric ratios as θ ranges from 0 to 90
degrees.
Special Right Triangles
Discovering Geometry 9.3 Two Special Right Triangles
Discovering Geometry 9.4 Story Problems
hypotenuse
θ
opposite of θ
Adjacent to θ
opposite
hypotenuse
sine of θ = sin θ =
hypotenuse
cosecant of θ = csc θ = opposite
adjacent
cosine of θ = cos θ = hypotenuse
hypotenuse
secant of θ = sec θ = adjacent
opposite
tangent of θ = tan θ = adjacent
adjacent
cotangent of θ = cot θ = opposite
Trigonometry
Discovering Geometry 12.1Trigonometric Ratios
Discovering Geometry 12.2 Problem Solving in Right
Triangles
Performance Tasks
Georgia CCGPS Analytic Geometry Unit 2: Right
Triangle Trigonometry
Geometric simulation software, applets, and graphing calculators can be used to explore the relationship
between sine and cosine.
Students may use graphing calculators or programs, tables, spreadsheets, or
computer algebra systems to solve right triangle problems.
Example:
• Find the height of a tree to the nearest tenth if the angle of elevation of the
sun is 28° and the shadow of the tree is 50 ft.
Unit 6
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
Evaluative Criteria/Assessment Level
Descriptors (ALDs):
Claim 1 Clusters:
Define trigonometric ratios and solve
problems involving right triangles
Claim 2 Clusters:
Define trigonometric ratios and solve
problems involving right triangles
Claim 3 Clusters:
Prove theorems involving similarity
Go here for Sample SBAC items
2015 – 2016
Stage 2 - Evidence
Sample Assessment Evidence
Concepts and Procedures
Level 3 students should be able to use the Pythagorean Theorem, trigonometric ratios, and the sine and cosine of
complementary angles to solve unfamiliar problems with minimal scaffolding involving right triangles, finding the
missing side or missing angle of a right triangle.
Level 4 students should be able to solve unfamiliar, complex, or multistep problems without scaffolding involving
right triangles
Problem Solving
Level 3 students should be able to map, display, and identify relationships, use appropriate tools strategically, and
apply mathematics accurately in everyday life, society, and the workplace. They should be able to interpret
information and results in the context of an unfamiliar situation.
Level 4 students should be able to analyze and interpret the context of an unfamiliar situation for problems of
increasing complexity and solve problems with optimal solutions.
Communicating Reasoning
Level 3 students should be able to use stated assumptions, definitions, and previously established results and
examples to test and support their reasoning or to identify, explain, and repair the flaw in an argument. Students
should be able to break an argument into cases to determine when the argument does or does not hold.
Level 4 students should be able to use stated assumptions, definitions, and previously established results to support
their reasoning or repair and explain the flaw in an argument. They should be able to construct a chain of logic to
justify or refute a proposition or conjecture and to determine the conditions under which an argument does or does
not apply.
Go here for more information about the Achievement Level Descriptors for Mathematics:
Unit 6
Clover Park School District 2015-2016
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Geometry Mathematics Curriculum Guide
2015 – 2016
Stage 3 – Learning Plan: Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
LEARNING ACTIVITIES:
NOTES:
Recommended to use EngageNY Geometry Module 2: Lessons 21,24-30
Include Performance Task: Example Trig Performance Tasks
Connection to Prior Grades:
8.G.6-8: These standards develop understanding
of Pythagorean Theorem and it converse. Also, the
Pythagorean Theorem is applied to develop
distance formula.
Daily Lesson Components
Learning Target
Warm-up
Activities
• Whole Group:
• Small Group/Guided/Collaborative/Independent:
• Whole Group:
Checking for Understanding (before, during and after):
Assessments
Unit 6
Clover Park School District 2015-2016
Algebra Review:
Solving for a variable in a formula
A-CED.4
Holt Algebra 1: Chapter 2, Lesson 5 Solving for a
variable.
Simplifying Radical Expressions:
N-RN.1,2
Holt Algebra 1: Chapter 11, Lesson 6 Radical
Expressions.
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