Standards:
GSRT.A.1 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.
GSRT.A.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.
GSRT.A.3 Use the properties of similarity
transformations to establish the AA criterion for two triangles to be
similar.
GSRT.B.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.
GSRT.B.5 Use
congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
GSRT.C.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.
GSRT.C.7 Explain
and use the relationship between the sine and cosine of complementary angles.
GSRT.C.8 Use
trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.^{★}
GMG.A.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).^{★}
GMG.A.2 Apply concepts of density based on area and volume in modeling
situations (e.g., persons per square mile, BTUs per cubic foot). ^{★}
GMG.A.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).^{
★}
GSRT.D.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.
GSRT.D.10 (+)Prove the Laws of Sines and Cosines and use them to solve problems.
GSRT.D.11 (+)Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and nonright triangles (e.g., surveying problems,
resultant forces).
Focus Standards of Mathematical Practice:
MP.1 Make
sense of problems and persevere in solving them.
MP.2 Reason
abstractly and quantitatively.
MP.3
Construct viable arguments and critique the reasoning of others.
MP.4 Model
with mathematics.
MP.5 Use
appropriate tools strategically.
MP.6 Attend
to precision.
MP.7 Look for
and make use of structure.
MP.8 Look for
and express regularity in repeated reasoning.
Instructional Outcomes:
Full
Development of the Major Clusters, Supporting Clusters, Additional Clusters
and Mathematical Practices for this unit could include the following
instructional outcomes:
GSRT.A.1
·
I can define image,
preimage, scale factor, center, and similar figures as they relate to
transformations
·
I can identify a
dilation stating its scale factor and center
·
I can verify
experimentally that a dilated image is similar to its preimage by showing
congruent corresponding angles and proportional sides
·
I can verify
experimentally that a dilation takes a line not passing through the center of
the dilation to a parallel line by showing the lines are parallel
·
I can verify
experimentally that dilation takes a line not passing through the center of
the dilation unchanged by showing that it is the same line
GSRT.A.2
·
I can, by using
similarity transformations, explain that triangles are similar if all pairs
of corresponding angles are congruent and all corresponding pairs of sides
are proportional
·
I can, given two
figures, decide if they are similar by using the definition of similarity in
terms of similarity transformations
GSRT.A.3
·
I can recall the
properties of similarity transformations
·
I can establish the
AA criterion for similarity of triangles by extending the properties of
similarity transformations to the general case of any two similar triangles
GSRT.B.4
·
I can recall
postulates, theorems, and definitions to prove theorems about triangles
·
I can prove theorems
involving similarity 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)
GSRT.B.5
·
I can recall
congruence and similarity criteria for triangles
·
I can use congruency
and similarity theorems for triangles to solve problems
·
I can use congruency
and similarity theorems for triangles to prove relationships in geometric
figures
GSRT.C.6
·
I can name the sides
of right triangles as related to an acute angle
·
I can recognize that
if two right triangles have a pair of acute, congruent angles that the
triangles are similar
·
I can compare common
ratios for similar right triangles and develop a relationship between the
ratio and the acute angle leading to the trigonometry ratios
GSRT.C.7
·
I can use the
relationship between the sine and cosine of complementary angles
·
I can identify the
sine and cosine of acute angles in right triangles
·
I can identify the
tangent of acute angles on right triangles
·
I can explain how the
sine and cosine of complementary angles are related to each other
GSRT.C.8
·
I can recognize which
methods could be used to solve right triangles in applied problems
·
I can solve for an
unknown angle or side of a right triangle using sine, cosine, and tangent
·
I can apply right
triangle trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems
GMG.A.1
·
I
can use measures and properties of geometric shapes to describe real world
objects
·
I
can, given a realworld object, classify the object as a known geometric
shape – use this to solve problems in context
GMG.A.2
·
I can define density
·
I can apply concepts
of density based on area and volume to model reallife situations (e.g.,
persons per square mile, BTUs per cubic foot)
GMG.A.3
·
I can describe a
typographical grid system
·
I can 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
system based on ratios)
(+)GSRT.D.9
·
I can recall right
triangle trigonometry to solve mathematical problems
·
I can apply the area
of a triangle formula by using the formula A=1/2ab sin(c) to solve real world
problems I can derive the formula A=1/2ab sin(c)for the area of a triangle by
drawing an auxiliary line from a vertex perpendicular to the opposite side
(+)GSRT.D.10
·
I can use the Laws of
Sines and Cosines to find missing angles or side length measurements
·
I can Prove the Law
of Sines
·
I can prove the Law
of Cosines
·
I can recognize when
the Law of Sines or Law of Cosines can be applied to a problem and solve
problems in context using them
(+)GSRT.D.11
·
I can determine from
given measurements in right and nonright triangles whether it is appropriate
to use the Law of Sines or Cosines
·
I can apply the Law
of Sines and the Law of Cosines to find unknown measurements in right and
nonright triangles (e.g., surveying problems, resultant forces)
