Download Marshall AbG Subject Geometry Honors Grade 10 Unit # 3 Pacing 6

Subject
Montclair Public Schools
CCSS Geometry Honors Unit: Marshall A.b.G
10
Unit # 3
Pacing
Geometry
Grade
Honors
Similarity, Trigonometry, and Transformations
6-8 weeks (MP 3)
Unit Name
Overview
Unit 3 takes previously learned geometric figures and transforms them (rotations, reflections, translations, dilations) throughout the
coordinate plane. Students should be familiar recognizing the transformations as well as perform indicated transformations. Based on
knowledge of dilations and other transformations, students will begin to understand similarity of figures. Students will reuse previous
information from congruence in Unit 2 to develop similarity statements and formal proofs for similar triangles. Students will use the idea
of similarity to explore what happens in the case of a right triangle and be able to find geometric mean. Right triangles will be studied in
depth. Pythagorean Theorem, trigonometric ratios, Law of Sines/Cosines and finding area of non-right triangles will be explored.
Standard #
Standard
MC, SLO
Student Learning Objectives
Depth of
SC,
#
Knowledge
or
AC
Prove theorems about right triangles.
1 Apply the following theorems about triangles:
Theorems include: a line parallel to one side
a) The Side-Side-Side Similarity Theorem
G.SRT.4
of a triangle divides the other two
MC
b) The Side-Angle-Side Similarity Theorem
Level 4
proportionally, and conversely; the
c) The Triangle Proportionality Theorem and its
Pythagorean Theorem proved using
converse.
similarity.
d) Triangle Midsegment Theorem
e) The Angle Bisector Theorem
1: Clarke
2
Explore similar triangle’s proportional perimeters
3
Prove the following theorems about right
triangles. When an altitude is drawn from the
right angle to the hypotenuse:
a) the two triangles formed are similar to the
given triangle, and to each other. The altitude
drawn is the geometric mean between the
measures of the two segments of the hypotenuse.
b) The sides are geometric means between the
measures of the hypotenuse and the segment of
the hypotenuse adjacent to it.
c) the Pythagorean Theorem and its converse.
4
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.
5
Prove the following theorems about special
triangles:
a) In a 45-45-90 triangle, the hypotenuse is sqrt2
times as long as the leg.
b) In a 30-60-90 triangle, the hypotenuse is twice
as long as the shorter leg, and the longer leg is
sqrt3 times as long as the shorter leg.
Define Sine, Cosine, Tangent as the ratio of side
lengths in right triangles, using the concepts of
similar triangles and scale factors.
Level 2
MC
G.SRT.7
Explain and use the relationship between
the sine and cosine of complementary
angles.
6
MC
G.SRT.8
2: Clarke
Use trigonometric ratios and the
Pythagorean Theorem to solve right
triangles in applied problems.
7
MC
Demonstrate using the definitions of the three
basic trigonometric ratios, that the Sine of one of
the acute angles in a right triangle references the
same side in the same ratio as the Cosine of the
other acute angle.
Construct and solve appropriate trigonometric
ratios in right triangles to find specific side lengths
Level 2
Level 4
or angle measures
8
Define angles of elevation and angles of
depressions
9
Construct right triangles and label them
appropriately, to represent relationships in a
descriptive scenario
10
G.SRT.9
G.SRT.10
Derive the formula A = ½ ab sin C for the
area of a triangle by drawing an auxiliary
line from a vertex perpendicular to the
opposite side
Prove the Laws of Sines and Cosines and
use them to solve problems.
11
SC
12
SC
Model and solve a series of descriptive scenarios,
using right triangle models
Derive the area of a triangle, using the formula
A = ½ bh and trigonometric ratios.
Level 3
Solve triangles using the Law of Sines/Cosines, or
a combination of the two laws.
Level 4
G.SRT.11
3: Clarke
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)
13
M
m
Model and solve a series of descriptive scenarios
and special projects requiring the application of
the Law of Sines and the Law of Cosines
Level 4
G.SRT.1
G.SRT.2
G.SRT.3
G.GPE.7
G.CO.2
4: Clarke
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. B. The
dilation of a line segment is longer or
shorter in the ratio given by the scale
factor.
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.
Use the properties of similarity
transformations to establish the AA
criterion for two triangles to be similar
Use coordinates to compute perimeters of
polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Represent transformations in the plane
using, e.g., transparencies and geometry
software; describe transformations as
functions that take points in the plane as
inputs and give other points as outputs.
14
SC
Level 3
15
Use the definition of similarity in terms of
similarity transformations to decide if two figures
are similar.
Level 2
16
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.
Level 4
17
Use the properties of similarity transformations to
establish the AA criterion for two triangles to be
similar.
MC
MC
18
MC
19
MC
Justify 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
b) The dilation of a line segment is longer or
shorter in the ratio given by the scale factor.
Use coordinates to compute perimeters of
polygons and areas of triangles and rectangles
and explore the relationship in regards to similar
figures
Develop and perform rigid transformations that
include reflections, rotations, translations, and
dilations using geometric software, graph paper,
tracing paper, and geometric tools and compare
them to non-rigid transformations.
Level 4
Level 4
Level 4
Compare transformations that preserve
distance and angle to those that do not
(e.g., translation versus horizontal stretch)
G.CO.3
Given a rectangle, parallelogram, trapezoid,
or regular polygon, describe the rotations
and reflections that carry it onto itself.
MC
20
Describe the transformations (rotations and
reflections) that would carry the given polygon
onto itself.
Develop definitions of rotations, reflections, MC
21
Apply the definitions of angles, circles, parallel
G.CO.4
and translations in terms of angles, circles,
lines, perpendicular lines and line segments to
perpendicular lines, parallel lines, and line
describe rotations, reflections, and translations.
segments.
Given a geometric figure and a rotation,
MC
22
Draw given geometric figures and their images
G.CO.5
reflection, or translation, draw the
using graph paper, tracing paper, or geometric
transformed figure using, e.g., graph paper,
software.
tracing paper, or geometry software.
Specify a sequence of transformations that
23
Identify a sequence of transformations that will
will carry a given figure onto another.
carry a given figure onto another.
Use geometric descriptions of rigid motions
24
Use rigid transformations to determine, explain
G.CO.6
to transform figures and to predict the
MC
the congruence of geometric figures.
effect of a given rigid motion on a given
figure; given two figures, use the definition
of congruence in terms of rigid motions to
decide if they are congruent.
Mathematical
Selected Opportunities for Connections to Mathematical Practices
Practice #
MP.1
MP.3
Make sense of problems and persevere in solving them.
Construct viable arguments and critique the reasoning of others.
MP.4
Model with mathematics
5: Clarke
Level 2
Level 3
Level 3
Level 4
MP.5
Use appropriate tools strategically.
Big Ideas
 Triangles can be proved similar by AA, ASA, SSS, SAS, AAS
 Similar figures have angles that are congruent and sides in proportion.
 Sin/Cos/Tan are ratios of the sides of the triangle, and can be used to solve for missing side lengths or angle measures.
 Transformations (reflections, rotations, and translations) will change a figure’s position, but not the size or shape. A dilation will
change the size, but not shape, of the figure.
Essential Questions
 How do you prove triangle similarity?
 What are the differences between similar and congruent figures?
 How is trigonometry used to find unknown measures?
 How do you change a figure’s position, without changing its size or shape?
 How you do change a figure’s size, without changing its shape?
Assessments
Required: Unit 3 Interim Assessment
Recommended: Quizzes – Transformations, Similarity (proportions), Trig Ratios (sin/cos/tan)
Tests – Transformations, Similarity (include finding perimeters), Trig Ratios (included angle of depression/elevation, word problems)
Key Vocabulary
 Transformation
 Similarity
 Sine
 Cosine
 Tangent
Suggested Resources (list specific chapters and or page numbers from existing text that correspond to the SLOs and Standards)
Chapter 7 Sections: 7-2, 7-3
6: Clarke
Prentice Hall
Geometry Textbook
Chapter 8 Sections: All
Chapter 9 Sections: 9-1, 9-2, 9-3, 9-5
Chapter 12 Sections: All
Learning Experiences (last area to be completed)
Instructional Focus
Student Learning Objectives

What can I do to make the work
maximally engaging and effective?

What content should we cover? What
content needs to be “uncovered”?

When should the “basics” come first?
When should they be on a “need to
know” basis?

When should I teach, when should I
coach, and when should I facilitate
student “discovery”?

How do I know who and where the
learners are?

In order to truly meet the standard,
what should they be able to do
independently (transfer)? What should
I be doing to make them more
independent and able to transfer?

What events will help students practice
& get feedback in transfer using the
learning in realistic ways?

What mathematical practices will my
students engage in to make
connections to the content?


7: Clarke
List SLOs that are addressed via
instructional focus
All SLOs should be addressed; if listed in
unit then they should be taught
Assessments


How will you assess these learning events?
What types of assessments will you use to check for
understanding?