Download Marshall Ab Subject Geometry Academic Grade 10 Unit # 3 Pacing

Subject
Geometry
Academic
Grade
Montclair Public Schools
CCSS Geometry Academic Unit: Marshall A.b.
10
Unit # 3
Pacing
8-10 weeks
Similarity, Trigonometry, & Transformations
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. 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. 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.4
1
2
3
4
5
6
MC
7
8
9
Apply the Side, Side, Side Similarity Theorem
Apply the Side, Angle, Side Similarity Theorem
Apply the Triangle Proportionality Theorem and its converse
Apply the Side Splitter Theorem
Apply the Angle Bisector Theorem
Apply the fact that similar triangles have corresponding
medians, altitudes, and angle bisectors that are
proportional
Apply the fact that similar triangles have proportional
perimeters
Apply the fact that when an altitude is drawn from the right
angle to the hypotenuse, the two triangles formed are
similar to the given triangle, and to each other
Apply the fact that the altitude drawn to the hypotenuse is
the geometric mean between the measures of the two
segments of the hypotenuse
4
4
4
4
4
4
4
4
4
4
10
11
12
13
G.SRT.6
G.SRT.7
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.
Explain and use the relationship between the
sine and cosine of complementary angles.
14
MC
15
MC
16
G.SRT.8
G.SRT.9
G.SRT.10
G.SRT.11
G.SRT.1
MC
Use trigonometric ratios and the Pythagorean
Theorem to solve right triangles in applied
problems.
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
Prove the Laws of Sines and Cosines and use
them to solve problems
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)
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
17
18
19
20
SC
SC
21
22
23
24
MC
25
SC
26
Apply the fact that each leg of the original right triangle is
the geometric mean between the measures of the
hypotenuse and the segment of the hypotenuse adjacent to
it
Apply the Pythagorean Theorem and its converse
Apply the relationship between the sides of a 45-45-90
triangle
Apply the relationships between the sides of 30-60-90 right
triangles
Apply the concepts of similar triangles and scale factors to
derive the Sine, Cosine and Tangent functions in terms of
acute angles of a right triangle
4
4
4
4
Prove, using the definitions of the three basic trigonometric
ratios, that the Sine of one of the acute angles in a right
triangle equals the Cosine of the other acute angle.
Construct and solve appropriate trigonometric ratios in right
triangles to find specific side lengths or angle measures
Differentiate angles of elevation from angles of depression
Construct right triangles and label them appropriately, to
represent relationships in a descriptive scenario
Model and solve a series of descriptive scenarios, using right
triangle models.
Apply right triangle trigonometry to derive the equation A =
1/2 ab sin(C) from the equation A=1/2bh
4
Apply the Law of Sines to solve triangles
Apply the Law of Cosines to solve triangles
Apply a combination of the Law of Sines and Law of Cosines
to solve real world problems
Apply the Law of Sines and Law of Cosines to solve a range
of real world problems
4
4
4
Apply the properties of dilations to show that a dilation
takes a line not passing through the center to a parallel line
Apply the properties of dilations to show that a dilation
leaves a line passing through the center unchanged
4
4
4
4
3
4
4
G.SRT.2
G.SRT.3
G.CO.2
G.CO.3
G.CO.4
G.CO.5
G.CO.6
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
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. Compare transformations that
preserve distance and angle to those that do not
(e.g., translation versus horizontal stretch)
Given a rectangle, parallelogram, trapezoid, or
regular polygon, describe the rotations and
reflections that carry it onto itself.
Develop definitions of rotations, reflections, and
translations in terms of angles, circles,
perpendicular lines, parallel lines, and line
segments
Given a geometric figure and a rotation,
reflection, or translation, draw the transformed
figure using, e.g., graph paper, tracing paper, or
geometry software. Specify a sequence of
transformations that will carry a given figure onto
another
Use geometric descriptions of rigid motions to
transform figures and to predict the 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
MC
27
Apply the properties of dilations to determine under what
conditions a line segment will become longer or shorter
28
Apply the definition of similarity in terms of similarity
transformations to decide if two figures 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.
4
30
Apply the properties of similarity transformations to
establish the AA criterion for two triangles to be similar.
4
31
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.
4
32
Investigate which transformations (rotations and
reflections) will carry the given polygon onto itself.
3
33
Apply the definitions of angles, circles, parallel
lines, perpendicular lines and line segments to describe
rotations, reflections, and translations.
4
34
Apply the properties of transformations to draw a given
geometric figure and its images using graph paper, tracing
paper or geometric software.
Identify a sequence of transformations that will carry a
given figure onto another.
4
Apply rigid transformations to determine, explain and
prove congruence of geometric figures
4
29
MC
MC
MC
MC
MC
35
36
MC
4
2
Big Ideas
Right triangles
Pythagorean Theorem
Right Triangle Trigonometry
Transformations of geometric shapes
Law of Sines and Cosines
Area of a triangle with Trigonometry
Essential Questions
How can the Pythagorean Theorem be used to classify a triangle?
What happens to a geometric figure under a reflection, translation, rotation, dilation and a combination of any of these?
How can trigonometry be used to find the area of a triangle?
How can right triangle trigonometry be used to solve right triangles?
How does the Pythagorean Theorem and Right Triangle Trigonometry apply to real world problems?
Key Vocabulary
Sine, Cosine and Tangent Functions
Rotation, reflection, translation and dilation
Solving a triangle
Suggested Resources (list specific chapters and or page numbers from existing text that correspond to the SLOs and Standards)
Chapter 9
Chapter 12