Download Geometry curriculum guide

COURSE CONTENT
MATHEMATICS
GEOMETRY
CCRS
QUALITY
CORE
EVIDENCE OF STUDENT
ATTAINMENT
CONTENT STANDARDS
RESOURCES
FIRST SIX WEEKS
1
C.1.a
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line
segment based on the undefined notions of point, line, distance along a line, and
distance around a circular arc. [G-CO.1]
Students:
Given undefined notions of point, line,
distance along a line, and distance
around a circular arc,


12
D.1.a
12. Make formal geometric constructions with a variety of tools and methods such as
compass and straightedge, string, reflective devices, paper folding, and dynamic
geometric software. Constructions include copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a given
line through a point not on the line. [G-CO.12]
Develop precise definitions of
angle, circle, perpendicular
line, parallel line, and line
segment,
Identify examples and nonexamples of angles, circles,
perpendicular lines, parallel
lines, and line segments.
Glencoe Geometry Text, p.17, 30, 39, 40, 55,
207, 222, 245, 266, 267, 273, 275, 323, 334,
494, 670-671, 718
Students:


Glencoe Geometry Text, Ch. 1, p.5-39 and
Ch. 10, p.694-714
Make and justify formal
geometric constructions with a
variety of tools and methods
(e.g., compass and
straightedge, string, reflective
devices, paper folding,
dynamic geometric software,
etc.) including the following:
Copying a segment; copying
an angle; bisecting a segment;
bisecting an angle;
constructing perpendicular
lines, including the
perpendicular bisector of a line
segment; and constructing a
line parallel to a given line
through a point not on the line,
Compare and contrast different
methods for doing the same
construction, and identify
1
COURSE CONTENT
MATHEMATICS
GEOMETRY
geometric properties that
justify steps in the
constructions.
13
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a
circle. [G-CO.13]


2
G.1.e
2. 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). [G-CO.2]
Glencoe Geometry Text, p.740
Students:
Use tools (e.g., compass,
straight edge, geometry
software) and geometric
relationships to construct
regular polygons inscribed in
circles,
Explain and justify the
sequence of steps taken to
complete the construction.
Students:
Given a variety of transformations
(translations, rotations, reflections, and
dilations),

Represent the transformations
in the plane using a variety of
methods (e.g., technology,
transparencies, semitransparent mirrors (MIRAs),
patty paper, compass),

Describe transformations as
functions that take points in the
plane as inputs and give other
points as outputs, explain why
this satisfies the definition of a
function, and adapt function
notation to that of a mapping
[e.g., f(x,y) → f(x+a, y+b)],
Glencoe Geometry Text, p. 296-302, 511-517,
623-631, 632-638, 639, 640-646, 650, 651659, 674-681
2
COURSE CONTENT
MATHEMATICS
GEOMETRY

3
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself. [G-CO.3]
Students:
Given a collection of figures that include
rectangles, parallelograms, trapezoids, or
regular polygons,




4
C.1.a
E.1.e
4. Develop definitions of rotations, reflections, and translations in terms of angles,
circles, perpendicular lines, parallel lines, and line segments. [G-CO.4]
Compare transformations that
preserve distance and angle to
those that do not.
Identify which figures that
have rotations or reflections
that carry the figure onto itself,
Perform and communicate
rotations and reflections that
map the object to itself,
Distinguish these
transformations from those
which do not carry the object
back to itself,
Describe the relationship of
these findings to symmetry.
Glencoe Geometry Text, p.623-631, 632-638,
640-646, 650, 651-659
Students:


Glencoe Geometry Text, p.663-669
Use geometric terminology
(angles, circles, perpendicular
lines, parallel lines, and line
segments) to describe the
series of steps necessary to
produce a rotation, reflection,
or translation,
Use these descriptions to
communicate precise
definitions of rotation,
reflection, and translation.
3
COURSE CONTENT
MATHEMATICS
GEOMETRY
5
E.1.a
E.1.e
5. 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.
[G-CO.5]
Students:
Given a geometric figure,


9
C.1.a
C.1.e
D.1.c
C.1.b
D.1.b
9. Prove theorems about lines and angles. Theorems include vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles are
congruent and corresponding angles are congruent; and points on a perpendicular
bisector of a line segment are exactly those equidistant from the segment's endpoints.
[G-CO.9]
Produce the image of the
figure under a rotation,
reflection, or translation using
graph paper, tracing paper, or
geometry software,
Describe and justify the
sequence of transformations
that will carry a given figure
onto another.
Glencoe Geometry Text, p. 144-150, 151-159,
180-186, 207-214, 324-333
Students:




Glencoe Geometry Text, p.294-295, 623-631,
632-638, 639, 640-646, 650, 651-659
Make, explain, and justify (or
refute) conjectures about
geometric relationships with
and without technology,
Explain the requirements of a
mathematical proof,
Present a complete
mathematical proof of
geometry theorems including
the following: vertical angles
are congruent; when a
transversal crosses parallel
lines, alternate interior angles
are congruent and
corresponding angles are
congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment's
endpoints,
Critique proposed proofs made
by others.
SECOND SIX WEEKS
4
COURSE CONTENT
MATHEMATICS
GEOMETRY
6
E.1.a
6. 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. [G-CO.6]
Students:
Given geometric descriptions of rigid
motions,



Glencoe Geometry Text, p. 294-295, 296-302,
623-631, 632-638, 640-646, 651-659, 682683
Predict the effect of the rigid
motion on a given figure,
Produce the image of a figure
under the transformation,
Compare and contrast the
predictions to the actual
transformation.
Given two figures,

7
C.1.g
E.1.b
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and corresponding
pairs of angles are congruent. [G-CO.7]
Determine if a sequence of
rotations, reflections, and
translations will carry the first
to the second, and if so justify
their congruence by the
definition of congruence in
terms of rigid motions.
Students:
Given a triangle and its image under a
sequence of rigid motions (translations,
reflections, and translations),

Glencoe Geometry Text, p. 255-263, 294-295,
296-302, 623-631, 632-638, 640-646, 651659, 682-683
Verify that corresponding sides
and corresponding angles are
congruent.
Given two triangles that have the same
side lengths and angle measures,

Find a sequence of rigid
motions that will map one onto
5
COURSE CONTENT
MATHEMATICS
GEOMETRY
the other.
8
10
C.1.f
C.1.e
D.2.b
D.2.j
8. Explain how the criteria for triangle congruence, angle-side-angle (ASA), sideangle-side (SAS), and side-side-side (SSS), follow from the definition of congruence
in terms of rigid motions. [G-CO.8]
Students:
10. Prove theorems about triangles. Theorems include measures of interior angles of
a triangle sum to 180o, base angles of isosceles triangles are congruent, the segment
joining midpoints of two sides of a triangle is parallel to the third side and half the
length, and the medians of a triangle meet at a point. [G-CO.10]
Students:





Glencoe Geometry Text, p. 296-302, 682-683
Use rigid motions and the
basic properties of rigid
motions (that they preserve
distance and angle), which are
assumed without proof to
establish that the usual triangle
congruence criteria make sense
and can then be used to prove
other theorems.
Make, explain, and justify (or
refute) conjectures about
geometric relationships with
and without technology,
Explain the requirements of a
mathematical proof,
Present a complete
mathematical proof of
geometry theorems about
triangles, including the
following: measures of interior
angles of a triangle sum to
180o; base angles of isosceles
triangles are congruent; the
segment joining midpoints of
two sides of a triangle is
parallel to the third side and
half the length; the medians of
a triangle meet at a point,
Critique proposed proofs made
by others.
Glencoe Geometry Text, p. 246-254, 255-263,
264-272, 275-282, 285-293, 303-309, 324333, 335-343, 344-351, 355-362, 371-380,
490-499, 546
6
COURSE CONTENT
MATHEMATICS
GEOMETRY
11
C.1.e
C.1.i
D.2.g
11. Prove theorems about parallelograms. Theorems include opposite sides are
congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each
other; and conversely, rectangles are parallelograms with congruent diagonals.
[G-CO.11]




14
14. Verify experimentally the properties of dilations given by a center and a scale
factor. [G-SRT.1]
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. [G-SRT.1.a]
Glencoe Geometry Text, p. 403-411, 413-421,
423-429, 431-438
Students:
Make, explain, and justify (or
refute) conjectures about
geometric relationships with
and without technology,
Explain the requirements of a
mathematical proof,
Present a complete
mathematical proof of
geometry theorems about
parallelograms, including the
following: opposite sides are
congruent, opposite angles are
congruent, the diagonals of a
parallelogram bisect each
other, and conversely,
rectangles are parallelograms
with congruent diagonals,
Critique proposed proofs made
by others.
Students:
Given a center of dilation, a scale factor,
and a polygonal image,

b.The dilation of a line segment is longer or shorter in the ratio given by the scale
factor. [G-SRT.1.b]


Glencoe Geometry Text, p. 672-673, 674-681
Create a new image by
extending a line segment from
the center of dilation through
each vertex of the original
figure by the scale factor to
find each new vertex,
Present a convincing argument
that line segments created by
the dilation are parallel to their
pre-images unless they pass
through the center of dilation,
in which case they remain on
the same line,
Find the ratio of the length of
the line segment from the
7
COURSE CONTENT
MATHEMATICS
GEOMETRY

15
C.1.h
E.1.c
E.1.d
15. 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.2]
center of dilation to each
vertex in the new image and
the corresponding segment in
the original image and
compare that ratio to the scale
factor,
Conjecture a generalization of
these results for all dilations.
Students:
Given two figures,

Glencoe Geometry Text, p. 469-477, 478-487,
511-517, 682-683
Determine if they are similar
by demonstrating whether one
figure can be obtained from the
other through a dilation and a
combination of translations,
reflections, and rotations.
Given a triangle,


Produce a similar triangle
through a dilation and a
combination of translations,
rotations, and reflections,
Demonstrate that a dilation and
a combination of translations,
reflections, and rotations
maintain the measure of each
angle in the triangles and all
corresponding pairs of sides of
the triangles are proportional.
THIRD SIX WEEKS
16
C.1.h
16. Use the properties of similarity transformations to establish the angle-angle (AA)
criterion for two triangles to be similar. [G-SRT.3
Students:
Given two triangles,

Glencoe Geometry Text, p. 478-487, 511-517,
682-683
Explain why if the measures of
8
COURSE CONTENT
MATHEMATICS
GEOMETRY

17
D.2.e
17. Prove theorems about triangles. Theorems include a line parallel to one side of a
triangle divides the other two proportionally, and conversely; and the Pythagorean
Theorem proved using triangle similarity. [G-SRT.4]
two angles from one triangle
are equal to the measures of
two angles from another
triangle, then measures of the
third angles must be equal to
each other,
Use this established fact and
the properties of a similarity
transformation to demonstrate
that the Angle-Angle (AA)
criterion for similar triangles is
sufficient.
Students:
Given a triangle and a line parallel to one
of the sides,

Glencoe Geometry Text, p. 478-487, 490-499,
501-508, 537-545
Prove the other two sides are
divided proportionally by
using AA, similarity
properties, previously proven
theorems and properties of
equality (Table 4).
Given a triangle with two of the sides
divided proportionally,

Prove the line dividing the
sides is parallel to the third
side of the triangle.
Given a right triangle,

Use similar triangles and
properties of equality (Table 4)
to prove the Pythagorean
Theorem.
9
COURSE CONTENT
MATHEMATICS
GEOMETRY
18
D.2.d
E.1.e
E.1.g
18. Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures. [G-SRT.5]
Students:
Given a contextual situation involving
triangles,


Glencoe Geometry Text, p. 255-263, 264-272,
273, 275-282, 283-284, 478-487, 490-499,
501-508, 511-517, 537-545
Determine solutions to
problems by applying
congruence and similarity
criteria for triangles to assist in
solving the problem,
Justify solutions and critique
the solutions of others.
Given a geometric figure,

19
H.1.b
19. 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.6]
Establish and justify
relationships in the figure
through the use of congruence
and similarity criteria for
triangles.
Students:
Given a collection of right triangles,



Glencoe Geometry Text, p. 558-566, 567,
568-577, 578
Construct similar right
triangles of various sizes for
each right triangle given,
Compare the ratios of the sides
of the original triangles to the
ratios of the sides of the similar
triangles,
Communicate observations
made about changes (or no
change) to such ratios as the
length of the side opposite an
angle to the hypotenuse, or the
side opposite the angle to the
side adjacent, as the size of the
angle changes or in the case of
10
COURSE CONTENT
MATHEMATICS
GEOMETRY

20
H.1.b
20. Explain and use the relationship between the sine and cosine of complementary
angles. [G-SRT.7]
similar triangles, remains the
same,
Summarize these observations
by defining the six
trigonometric ratios.
Students:
Given a right triangle,


Glencoe Geometry Text, p. 568-577
Explain why the two smallest
angles must be complements,
Compare the side ratios of
opposite/hypotenuse and
adjacent/hypotenuse for each
of these angles and discuss
conclusions.
Given a contextual situation involving
right triangles,

21
D.2.e
H.1.b
H.1.c
H.1.a
21. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems. [G-SRT.8]
Compare solutions to the
situation using the sine of the
given angle and the cosine of
its complement.
Students:
Given a contextual situation involving
right triangles,




Glencoe Geometry Text, p. 547-555, 568-577,
580-587, 588-597
Create a drawing to model the
situation,
Find the missing sides and/or
angles using trigonometric
ratios,
Find the missing sides using
the Pythagorean Theorem,
Use the above information to
interpret results in the context
11
COURSE CONTENT
MATHEMATICS
GEOMETRY
of the situation.
22
D.2.f
22. (+) Prove the Law of Sines and the Law of Cosines and use them to solve
problems. [G-SRT.10]
Students:
Given any triangle,


23
23. (+) 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). [G-SRT.11]
Derive the Law of Sines and
the Law of Cosines.
Use the Law of Sines or the
Law of Cosines to find
unknown lengths of sides and
measures of angles.
Students:
Given a contextual situation,

Glencoe Geometry Text, p. 588-597
Glencoe Geometry Text, p. 588-597, 598
Choose the appropriate law
and apply it to determine the
measures of unknown
quantities.
FOURTH SIX WEEKS
34
G.1.c
34. Determine areas and perimeters of regular polygons, including inscribed or
circumscribed polygons, given the coordinates of vertices or other characteristics.
[G-GMD.5]
Students:
Given the vertices of a regular polygon.




Glencoe Geometry Text, p. 807-817
Find the area.
Find the perimeter.
Find the area of the inscribed
or circumscribed polygon.
Find the perimeter of the
inscribed or circumscribed
polygon.
Given the apothem, perimeter, area, or
measurement of a side,
12
COURSE CONTENT
MATHEMATICS
GEOMETRY




35
D.4.a
F.2.a
F.2.b
E.1.f
E.1.h
35. Give an informal argument for the formulas for the circumference of a circle; area
of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments,
Cavalieri's principle, and informal limit arguments. [G-GMD1]
Find the area.
Find the perimeter.
Find the area of the inscribed
or circumscribed polygon.
Find the perimeter of the
inscribed or circumscribed
polygon.
Students:
Given a circle,



Glencoe Geometry Text, p. 697-705, 798-804,
863-870, 873-879, 880-887
Use repeated reasoning from
multiple examples of the ratio
of circle circumference to the
diameter, to informally
conjecture that the
circumference of any circle is a
little more than three times the
diameter,
Divide the circle into an equal
number of sectors, and
rearrange the sectors to form a
shape that is approaching a
parallelogram,
Make conjectures about the
area and perimeter of the new
shape as the number of sectors
becomes larger, and relate
those conjectures to the
original circle.
Given a cylinder,

Explain how a cylinder could
be divided into an infinite
number of circles, and the area
of those circles multiplied by
the height is the volume of the
13
COURSE CONTENT
MATHEMATICS
GEOMETRY
cylinder, and use Cavalieri's
Principle to demonstrate that if
two solids have the same
height and the same crosssectional area at every level,
then they have the same
volume.
Given a pyramid or cone,

36
D.4.a
F.2.a
F.2.c
36. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems. [G-GMD3]
Students:
Given a contextual situation that requires
finding the volume of a cylinder,
pyramid, cone, or sphere as part of its
solution,



37
D.4.a
F.2.a
F.2.c
37. Determine the relationship between surface areas of similar figures and volumes
of similar figures. (Alabama) [G-GMD.6]
Explain that the shapes could
be divided into cross-sections,
and the area of the crosssections is decreasing as the
cross-sections become further
away from the base, and the
area of an infinite number of
cross-sections is the volume of
a pyramid or cone.
Glencoe Geometry Text, p. 67-74, 863-870,
873-879, 880-887
Use an appropriate shape or 2D drawing to model the
situation,
Solve using the appropriate
formula,
Justify and explain the solution
and solution path in the context
of the given situation.
Students:
Given similar figures,
Glencoe Geometry Text, p. 896-902
14
COURSE CONTENT
MATHEMATICS
GEOMETRY

38
D.4.b
38. Identify the shapes of two-dimensional cross-sections of three-dimensional
objects, and identify three-dimensional objects generated by rotations of twodimensional objects. [G-GMD4]
Determine the relationship
between surface areas and
volumes of similar figures.
Students:
Given 3-D objects,




Glencoe Geometry Text, p. 647-648, 839-844
Conjecture about the
characteristics of geometric
shapes formed if a crosssection of a 3-D shape is taken,
Take 2-D cross-sections at
different angles of cut,
Explain the shape formed by
taking 2-D cross-sections,
Compare and contrast the
figures formed when the angle
of the cut changes.
Given 2-D objects,



39
D.1.a
E.1.h
39. 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.1]
Conjecture about the
characteristics of geometric
shapes formed from rotating a
2-D shape about a line,
Rotate the object about given
lines,
Explain the 3-D objects formed
if the 2-D object is rotated
about a line.
Students:
Given a real-world object,

Glencoe Geometry Text, throughout the text;
examples: p. 13, 75-77, 393-401, 818-824,
854-86
Select an appropriate
geometric shape to model the
15
COURSE CONTENT
MATHEMATICS
GEOMETRY


40
40. Apply concepts of density based on area and volume in modeling situations (e.g.,
persons per square mile, British Thermal Units (BTUs) per cubic foot).* [G-MG2]
Students:
Given a contextual situation involving
density,



41
41. 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).* [G-MG3]
object,
Provide a description of the
object through the measures
and properties of the geometric
shape which is modeling the
object,
Explain and justify the model
which was selected.
Model the situation by creating
an average per unit of area or
unit of volume,
Generate questions raised by
the model and defend answers
they produce to the generated
questions (e.g., should
population density be given per
square mile or per acre? What
insights might one yield over
the other?),
Explain and justify the model
in terms of the original context.
Students:
Given a contextual situation involving
design problems,


Glencoe Geometry Text, p. 797, 863-870,
873-879
Glencoe Geometry Text, various pages(see p.
T34 for a list)
Create a geometric method to
model the situation and solve
the problem,
Explain and justify the model
which was created to solve the
problem.
16
COURSE CONTENT
MATHEMATICS
GEOMETRY
FIFTH SIX WEEKS
24
24. Prove that all circles are similar. [G-C.1]
Students:
Given a collection of circles,



25
D.3.a
D.3.b
D.3.c
25. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles;
inscribed angles on a diameter are right angles; the radius of a circle is perpendicular
to the tangent where the radius intersects the circle. [G-C.2]
Show that in each case there
exists a transformation that
consists of a dilation and a
combination of rigid motions
that will take one of the circles
to any of the others,
Verify that the ratio of the
circumference of the circles
created through dilations is
equal to the ratio of the radii of
the circles, which is the same
as the scale factor of the
dilation,
Explain with logical reasoning
how a circle is fully defined by
a single parameter "r" so the
only non-translational changes
that can be made is alteration
of "r", which changes the size
and not the shape and therefore
the circles are similar.
Students:
Given circles with two points on the
circle,

Glencoe Geometry Text, p. 697-705
Glencoe Geometry Text, p. 697-705, 706714, 715-722, 723-730, 732-739
Compare the measures of the
angles (with and without
technology) formed by
creating radii to the given
points, creating chords from a
third point on the circle to the
given points, and creating
tangents from a third point
outside the circle to the given
points, and conjecture about
possible relationships among
17
COURSE CONTENT
MATHEMATICS
GEOMETRY

the angles,
Use logical reasoning to justify
(or deny) the conjectures (in
particular justify that an
inscribed angle is one half the
central angle cutting off the
same arc, and the
circumscribed angle cutting off
that arc is supplementary to the
central angle relating all three).
Given circles with chords from a point
on the circle to the endpoints of a
diameter,


Find the measure of the angles
(with and without technology),
conjecture about and explain
possible relationships,
Use logical reasoning to justify
(or deny) the conjectures (in
particular justify that an
inscribed angle on a diameter
is a right angle).
Given a circle with a tangent and radius
intersecting at a point on the circle,


Find the measure of the angle
at the intersection point (with
and without technology),
conjecture about and explain
possible relationships,
Use logical reasoning to justify
(or deny) the conjectures (in
particular justify that the radius
of a circle is perpendicular to
the tangent where the radius
intersects the circle.
18
COURSE CONTENT
MATHEMATICS
GEOMETRY
26
D.2.b
D.3.d
26. Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle. [G-C3]
Students:
Given a triangle,


Glencoe Geometry Text, p. 723-730, 740
Use tools (e.g., compass,
straight edge, geometry
software) to construct
inscribed and circumscribed
circles,
Explain and justify the
sequence of steps taken to
complete the construction.
Given a quadrilateral inscribed in a
circle,

27
D.3.c
27. (+) Construct a tangent line from a point outside a given circle to the circle. [GC4]
Students:
Given a circle and an external point,

28
F.1.d
F.1.e
28. Derive, using similarity, the fact that the length of the arc intercepted by an angle
is proportional to the radius, and define the radian measure of the angle as the constant
of proportionality; derive the formula for the area of a sector. [G-C5]
Conjecture possible
relationships among the angles
through the use of inscribed
angles use logical reasoning to
justify (or deny) the
conjectures (in particular,
justify that diagonally opposite
angles of a quadrilateral are
supplementary).
Use tools to construct a tangent
line to the circle from the
point.
Students:
Given an arc intercepted by an angle,

Glencoe Geometry Text, p. 732-739
Glencoe Geometry Text, p. 706-714, 798-804
Use dilations to create arcs
intercepted by the same central
19
COURSE CONTENT
MATHEMATICS
GEOMETRY



29
G.1.d
D.1.e
29. Derive the equation of a circle of given center and radius using the Pythagorean
Theorem; complete the square to find the center and radius of a circle given by an
equation. [G-GPE1]
angle with radii of various
sizes (including using dynamic
geometry software), and use
the ratios of the arc lengths and
radii to make conjectures
regarding possible relationship
between the arc length and the
radius,
Justify the conjecture for the
formula for any arc length (i.e.,
since 2πr is the circumference
of the whole circle, a piece of
the circle is reduced by the
ratio of the arc angle to a full
angle (360)),
Find the ratio of the arc length
to the radius of each
intercepted arc and use the
ratio to name the angle calling
this the radian measure of the
angle by extending the
definition of one radian as the
angle which intercepts an arc
of the same length as the
radius,
Develop the formula for the
area of a sector by interpreting
a circle as a complete
revolution and a sector as a
fractional part of a revolution.
Students:
Given the center (h,k) and radius (r) of a
circle,


Glencoe Geometry Text, p. 757-763
Explain and justify that every
point on the circle is a
combination of a horizontal
and vertical shift from the
center with a length equal to
the radius,
Create a right triangle from the
20
COURSE CONTENT
MATHEMATICS
GEOMETRY
center of a circle to a general
point on the circle, and show
that the legs of the right
triangle are the absolute values
of x-h and y-k, and the
hypotenuse is r, then apply
Pythagorean theorem to show
that r2 = (x - h)2 + (y - k)2.
Given an equation of a circle in general
form,

Complete the square to rewrite
the equation in the form r2 = (x
- h)2 + (y - k)2 and determine
the center and radius.
SIXTH SIX WEEKS
30
G.1.b
G.1.c
30. Use coordinates to prove simple geometric theorems algebraically. [G-GPE4]
Example: Prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1,√3) lies on the
circle centered at the origin and containing point (0,2).
Students:
Given coordinates and geometric
theorems and statements defined on a
coordinate system,

31
C.1.d
D.1.f
G.1.a
G.1.c
31. 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-GPE5]
Use the coordinate system and
logical reasoning to justify (or
deny) the statement or
theorem, and to critique
arguments presented by others.
Students:
Given a line,


Glencoe Geometry Text, p. 303-309, 403-411,
413-421, 423-429, 430-438, 439-448, 757763
Glencoe Geometry Text, p. 187, 188-196,
198-205, 206, 488-489
Create lines parallel to the
given line and compare the
slopes of parallel lines by
examining the rise/run ratio of
each line,
Create lines perpendicular to
21
COURSE CONTENT
MATHEMATICS
GEOMETRY

the given line by rotating the
line 90 degrees and compare
the slopes by examining the
rise/run ratio of each line,
Use understandings of similar
triangles and logical reasoning
to prove that parallel lines have
equal slopes and the slopes of
perpendicular lines are
negative reciprocals.
Given a geometric problem involving
parallel or perpendicular lines,

32
G.1.c
32. Find the point on a directed line segment between two given points that partitions
the segment in a given ratio. [G-GPE6]
Apply the appropriate slope
criteria to solve the problem
and justify the solution
including finding equations of
lines parallel or perpendicular
to a given line.
Students:
Given two points and a ratio that
partitions the segment between the
points,



Glencoe Geometry Text, p. 25-35, 490-499,
600-608, 674-681, 757-763
Construct a circle using one of
the given points as the center
and the distance between the
points as the radius,
Construct a dilation of the
circle using the given ratio as
the scale factor and find the
intersection between the
dilation and the equation of the
line passing through the given
points,
Justify and explain the reasons
for each step in the process of
finding a point that partitions a
22
COURSE CONTENT
MATHEMATICS
GEOMETRY
segment in a given ratio.
33
F.2.a
G.1.c
33. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.* [G-GPE7]
Students:
Given a contextual situation that requires
the perimeter and/or area of a polygon as
part of its solution,

42
F.1.c
42. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random
number generator). [S-MD6]
43
F.1.c
43. (+) Analyze decisions and strategies using probability concepts (e.g., product
testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7]
Find the solution to the
situation through the use of
coordinates and the distance
formula as appropriate,
through modeling the situation
in a Cartesian coordinate
system and explain and justify
the solution.
Students:
Given a contextual situation in which a
decision needs to be made,

Glencoe Geometry Text, p. P8-P9, 939-946
Use a random probability
selection model to produce
unbiased decisions.
Students:
Given a contextual situation in which a
decision needs to be made,

Glencoe Geometry Text, p. 56-64, 779-786
Glencoe Geometry Text, p. P8-P9, 931-937,
947-953
Use probability concepts to
analyze, justify, and make
objective decisions.
23