Download Unit 5 Similarity and Trigonometry Geometry

Unit 5 Similarity and Trigonometry
Geometry
Unit Goals – Stage 1
Number of Days: 34 days
2/21/17 – 4/7/17
Unit Description: In Unit 5, students are introduced to trigonometry through a revisit to similarity. Unlike congruence, here triangles are shown to be
similar with sides and angles being proportional, not congruent. Previously seen in middle school, the students will use the Pythagorean Theorem and
their knowledge of similar triangles to investigate the relationships found in the special right triangles. Using these relationships, students will solve for
parts of right triangles and apply these skills to real-life situations.
Materials: Patty paper, straight edges, compasses, calculators, dynamic software (Desmos, Geogebra)
Standards for Mathematical
Transfer Goals
Practice
Students will be able to independently use their learning to…
SMP 1 Make sense of problems
• Make sense of never-before-seen problems and persevere in solving them.
and persevere in solving
• Construct viable arguments and critique the reasoning of others.
them.
SMP 2 Reason abstractly and
Making Meaning
quantitatively.
UNDERSTANDINGS
ESSENTIAL
SMP 3 Construct viable
QUESTIONS
Students will understand that…
arguments and critique
Students will keep
• Ratios and proportions are necessary to the study of similarity.
the reasoning of others.
considering…
• The properties of similarity hold for all polygons.
SMP 4 Model with mathematics.
• How can you use
• Dilation can be used to show that figures are similar.
SMP 5 Use appropriate tools
ratios and
• In similar figures, all angles are congruent and corresponding side lengths are
strategically.
proportions to prove
proportional.
SMP 6 Attend to precision.
that triangles are
• Using similarity, one can solve for missing side lengths and angle measures in
SMP 7 Look for and make use of
similar?
applied situations.
structure.
•
How does knowing
• Trigonometric functions are a ratio of a set of given sides. If the angle is held
SMP 8 Look for and express
that triangles are
constant, a trigonometric ratio will remain the same, no matter the size of the triangle.
regularity in repeated
similar lead us to
• Sine, cosine and tangent can be used to solve for side lengths and angle measures of
reasoning.
trigonometry?
right triangles.
Standards for Mathematical
Content Clusters Addressed
[m] G-SRT.A Understand
similarity in terms
of similarity
transformations.
[m] G-SRT.B Prove theorems
involving
similarity.
Acquisition
SKILLS
Students will be skilled at and/or be able to…
• Use ratios to determine if figures are similar.
• Prove whether or not corresponding sides or angles are congruent.
• Use dilations to show that figures are similar.
• Apply AA, SSS, and SAS to prove that triangles are similar.
• Identify the opposite side, adjacent side, and hypotenuse of a right
triangle given a specified angle.
• Solve for a missing side using the Triangle Proportionality Theorem.
KNOWLEDGE
Students will know…
• The properties and vocabulary of
dilation.
• All circles are similar.
• Triangle similarity shortcuts: AA,
SSS, and SAS.
• The Triangle Proportionality
Theorem.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
1
Reposted 2/3/17
Unit 5 Similarity and Trigonometry
Geometry
Unit Goals – Stage 1
[m] G-SRT.C Define
trigonometric
ratios and solve
problems
involving right
triangles.
• Sine, Cosine and Tangent are
defined ratios that can be used to
solve for a missing measure in any
right triangle.
• The definitions of: complementary
angles, angles of elevation and
angles of depression.
• Use the Triangle Proportionality Theorem to prove that sides are
parallel to each other.
• Apply knowledge of similar triangles and rectangles to solve
problems.
• Use the coordinate plane to represent dilations.
• Use similarity criteria for triangles to solve for missing sides and to
prove relationships in geometric figures.
• Prove the Pythagorean Theorem using triangle similarity.
• Construct a tangent, sine or cosine ratio to solve for a missing side or
angle of a right triangle.
• Solve for an angle, when given two sides of a right triangle, by using
an inverse trigonometric ratio.
[m] G-GPE.B Use coordinates
to prove simple
geometric
theorems
algebraically.
[m] G-MG.A
Apply geometric
concepts in
modeling
situations.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
2
Reposted 2/3/17
Unit 5 Similarity and Trigonometry
Geometry
Assessed Grade Level Standards
Standards for Mathematical Practice
SMP 1
SMP 2
SMP 3
SMP 4
SMP 5
SMP 6
SMP 7
SMP 8
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Standards for Mathematical Content
[m] G-SRT.A Understand similarity in terms of similarity transformations.
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.
[m] G-SRT.B 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.
[m] G-SRT.C 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.
G-SRT.8.1 Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90° and 45°, 45°, 90°). CA
[m] G-GPE.B Use coordinates to prove simple geometric theorems algebraically. [Include distance formula; relate to Pythagorean theorem]
G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line
parallel or perpendicular to a given line that passes through a given point).
G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
[m] G-MG.A
Apply geometric concepts in modeling situations.
G-MG.1
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).
G-MG.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize
cost; working with typographic grid systems based on ratios).
Key: [m] = major clusters; [s] = supporting clusters, [a] = additional clusters

Indicates a modeling standard linking mathematics to everyday life, work, and decision-making
CA Indicates a California-only standard
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
3
Reposted 2/3/17
Unit 5 Similarity and Trigonometry
Geometry
Evidence of Learning – Stage 2
Assessment Evidence
Unit Assessment
Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency.
Concepts and skills that may be assessed in Claim1:
[m] G-SRT.A
• Students will use the definition of similarity in terms of similarity transformations to decide if two triangles are similar.
• Students will explain the meaning of similarity for triangles as the proportionality of all corresponding pairs of sides using similarity transformations.
• Students will use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
[m] G-SRT.B
• Students will use similarity to prove theorems about triangles.
• Students will use similarity criteria for triangles to solve problems.
• Students will use similarity criteria to prove relationships in geometric figures.
[m] G-SRT.C
• Students will use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
• Students will show, that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to the definitions of
trigonometric ratios for acute angles.
• Students will explain and use the relationship between the sine and cosine of complementary angles.
[m] G-GPE.B
• Students will use the slope criteria to solve geometric problems.
• Students will find the point on a directed line segment between two given points that partition the segment in a given ratio.
[m] G-MG.A
• Students will use geometric shapes, measures and properties to solve real-world problems.
• Students will apply geometric methods to solve design problems.
Claim 2: Students can solve a range of wellposed problems in pure and applied
mathematics, making productive use of
knowledge and problem-solving strategies.
Standard clusters that may be assessed in
Claim 2:
•
G-SRT.C
Claim 3: The student can clearly and precisely
construct viable arguments to support their own
reasoning and critique the reasoning of others.
Standard clusters that may be assessed in
Claim 3:
•
G-SRT.A
•
G-SRT.B
Claim 4: The student can analyze complex,
real-world scenarios and can construct and use
mathematical models to interpret and solve
problems.
Standard clusters that may be assessed in
Claim 4:
•
G-MG.A
Other Evidence
Formative Assessment Opportunities
• Opening Tasks
• Tasks
• Informal teacher observations
• Formative Assessment Lessons (FAL)
• Checking for understanding using active participation strategies
• Quizzes / Chapter Tests
• Exit slips/summaries
• Big Ideas Math Performance Tasks
• Modeling Lessons (SMP 4)
• SBAC Interim Assessment Blocks
Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
4
Reposted 2/3/17
Unit 5 Similarity and Trigonometry
Geometry
Learning Plan – Stage 3
Days
2-3
days
Learning
Target
I will solve for
the height of a
building, or
other tall
object, using
similar
triangles.
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Geometry
Expectations
(Activities and
Lessons)
Opening Task – How Tall Is That “Building”?
Students will use ratios and proportions to solve for the heights of
buildings on campus. This is a review of previous learning about ratios
and proportions, and a lead-in to the new idea of similar triangles and
proportions. Later in this unit, students will be able to use trigonometry to
repeat this activity via the tangent ratio and an inclinometer.
Curriculum Intranet
Applications:
• How Tall Is That
Building?
This task is a gateway into the entire unit on similar triangles and
trigonometry.
I investigate
polygon
similarity by…
2-4
days
•
•
•
•
•
•
•
Using similarity statements.
Finding corresponding lengths in similar polygons.
Finding scale factors.
Finding perimeters and areas of similar polygons.
Deciding whether polygons are similar.
Using similarity theorems to solve real-life problems.
Answering questions such as…
o How are a congruency transformation and a similarity transformation
alike? Different?
o Are congruent figures similar?
o Is there an advantage to simplifying a ratio in a proportion?
o If polygons have corresponding sides that are proportional, are the
polygons similar?
• Section 8.1
• STEM Video: Scale
Model of a Pool
Conceptual
Understanding:
• Illustrative
Mathematics:
Similarity
• Illustrative
Mathematics: Similar
Quadrilaterals
• Which One Doesn’t
Belong: Similarity
Procedural Skills and
Fluency:
• MathOpenRef:
Similar Triangles and
Polygons
Applications:
• STEM Video
Performance Task:
Pool Maintenance
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
5
Reposted 2/3/17
Unit 5 Similarity and Trigonometry
Geometry
Learning Plan – Stage 3
Days
Learning
Target
I will prove
triangles
similar by…
4-5
days
I will
investigate
proportionality
theorems
by…
2-3
days
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Geometry
Expectations
(Activities and
Lessons)
• Section 8.2
• Using the Angle-Angle Similarity Theorem to solve problems.
• Section 8.3
• Using the Side-Side-Side Similarity Theorem to solve problems.
•
•
•
•
Using the Side-Angle-Side Similarity Theorem to solve problems..
Using similarity theorems to solve real-life problems.
Proving slope criteria using similar triangles.
Answering questions such as…
o Can you assume that corresponding sides and corresponding angles
of any two similar triangles are congruent?
o What is the minimum amount of information you need about two
triangles to claim that they are similar?
o What are ways to use corresponding sides of two triangles to
determine that triangles are similar?
o Why isn’t there a SSSS Similarity Theorem for quadrilaterals?
o What are the key steps in proving the SSS Similarity Theorem? The
SAS Similarity Theorem?
o Why is there no Side-Side-Angle Similarity Theorem?
• Using the Triangle Proportionality Theorem and its converse to solve
problems.
• Using other proportionality theorems to solve problems.
• Answering questions such as…
o What are methods you can use to construct a series of parallel
segments?
o How is the Triangle Midsegment Theorem related to the Triangle
Proportionality Theorem?
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
6
Curriculum Intranet
Conceptual
Understanding:
• Geogebra Applet: AA
Similarity
• Geogebra Applet:
Triangle Similarity
Shortcuts
Procedural Skills and
Fluency:
• MathOpenRef:
Similar Triangles and
Polygons
• Section 8.4
Conceptual
Understanding:
• Illustrative
Mathematics: Finding
Triangle Coordinates
• Geogebra Applet:
Proportionality
Theorems
Reposted 2/3/17
Unit 5 Similarity and Trigonometry
Geometry
Learning Plan – Stage 3
Days
Learning
Target
I will
investigate
the
relationships
in right
triangles by…
•
5-8
days
I will solve for
parts of a right
triangle by…
5-6
days
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Geometry
Expectations
(Activities and
Lessons)
• Section 9.1
Using the Pythagorean Theorem.
• Section 9.2
Using the Converse of the Pythagorean Theorem.
• Section 9.3
Classifying triangles.
• STEM Video:
Finding the side lengths in special right triangles.
Height of a Rock
Solving real-life problems using special right triangles.
Wall
Identifying similar triangles.
Using geometric means.
Solving real-life problems using special right triangles and similar
triangles.
Answering questions such as:
o How would YOU prove the Pythagorean Theorem?
If p, q, and r satisfy the Pythagorean Theorem, will
o
o
o
o
o
o
Is the converse of every true statement true?
What are techniques to simplify radical expressions?
What is the difference between opposite and adjacent?
In your own words, what is the geometric mean?
How does a geometric mean apply to the real world?
Illustrate a situation that involves angle of elevation. Where is the
right angle in this situation?
Using the tangent, sine and cosine ratios.
Solving real-life problems involving the tangent, sine and cosine ratios.
Finding the sine and cosine angle measures in special right triangles.
Using inverse trigonometric ratios.
Solving right triangles.
Defining the trigonometric ratios for the acute angles in a right triangle.
Answering questions such as:
o How does the tangent of an acute angle in a right triangle change
as the angle measure increases?
o Why is it not possible to find the tangent of a right triangle or an
obtuse angle?
o How do you know which side of the triangle is “opposite”?
“Adjacent”?
7
Conceptual
Understanding:
• Geogebra Applet:
Special Right
Triangles
Applications:
• STEM Video
Performance Task:
Challenging the Rock
Wall
p q
r
, , and ?
2 2
2
o
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Curriculum Intranet
Section 9.4
Section 9.5
Section 9.6
Conceptual
Understanding:
• Illustrative
Mathematics:
Defining
Trigonometric Ratios
• Illustrative
Mathematics:
Tangent of Acute
Angles
• Geogebra Applet:
Ratio of Sides of a
Right Triangle
Reposted 2/3/17
Unit 5 Similarity and Trigonometry
Geometry
Learning Plan – Stage 3
Days
Learning
Target
o
o
o
o
o
o
o
o
o
2-3
days
2-3
days
I will check
my ability to
use similar
triangles to
solve real
world
problems by
participating
in the FAL.
I will prepare
for the unit
assessment
on similarity
and
trigonometry
by...
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Geometry
Expectations
(Activities and
Lessons)
How are the sine and cosine ratios the same? Different?
Will the sine and cosine ratios ever be greater than 1?
After solving for a missing side length, how can you check to see if
your answer is reasonable?
What do you have to know about a triangle to use the sine ratio or
the cosine ratio?
When you are solving a problem, how do you know whether to
use the sine, cosine or tangent ratio?
How would you use trigonometric ratios to find the missing angle
measures in a triangle?
How can you find the trigonometric ratios without a calculator?
Why are trig ratios important? Where/when can they be used?
Explain the difference between using the calculator to solve for
the trig ratio and using the calculator to solve for the inverse trig
ratio.
Curriculum Intranet
• Which One Doesn’t
Belong: Right
Triangle
Trigonometry
Procedural Skills and
Fluency:
• MathOpenRef:
Trigonometry
Functions
Applications:
• 3-Act Lesson: Boat
on a River
• FAL: Deducing
Relationships –
Floodlight Shadows
FORMATIVE ASSESSMENT LESSON
Deducing Relationships – Floodlight Shadows
Incorporating the Standards for Mathematical Practice (SMPs) along with
the content standards to review the unit.
1-2
days
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Unit Assessment (LBUSD Math Intranet, Assessment)
8
Reposted 2/3/17