Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
Designated Grading Period: 5th
Days to teach: 16
Assessment
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.(5) Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:
G.5(A) investigate
Investigate patterns to make
Given Circle M
Chord
Patterns should include
Big Ideas Geometry
patterns to make
conjectures about
Arc
numeric and geometric
10.3, 10.4, 10.5, 10.6
conjectures about
Diameter
properties.
 geometric
geometric relationships,
Inscribed angle
relationships including
including angles formed
Intercepted arc
Students should investigate
special segments
by parallel lines cut by a
Subtend
geometric relationships.
 angles of circles
transversal, criteria
Inscribed polygon
choosing from a
required for triangle
Circumscribed
Include some relationships
variety of tools.
congruence, special
circle
with algebraic expressions
segments of triangles,
Tangent
representing properties.
diagonals of
Secant
quadrilaterals, interior and
Segments of a
exterior angles of
chord
polygons, and special
Tangent segment
segments and angles of
Secant segment
circles choosing from a
External segment
Find
variety of tools.
Readiness Standard
a. mJ
Misconceptions:
b. mKIJ
 The student may make a conjecture based on limited investigation of
c. mJKI
patterns.
d. mIHE
 The student may randomly state a conjecture without investigating and
e. mGIF
recognizing patterns.
 The student may not know how to use a construction to make a
f. mFHE
conjecture.
Answers: a. 35°, b. 25, c.
 The student may not be able to perform constructions correctly.
120°, d. 62.5°, e. 20°, f.
117.5°
 The student may not state a conjecture using precise geometric
vocabulary.
2016-2017
Page 1
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
Designated Grading Period: 5th
Days to teach: 16
Assessment
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.(9) Similarity, proof, and trigonometry. The student uses the process skills to understand and apply relationships in right triangles. The student is expected to:
G.9(A) determine the
Determine the
Trigonometry
Use SOH-CAH-TOA to
Big Ideas Geometry
lengths of sides and
Opposite side
help students remember the
9.4, 9.5, 9.6
 lengths of sides
measures of angles in a
Opposite angle
trig ratios
 measures of angles
right triangle by applying
Right triangle
www.khanacademy.org
in a right triangle by applying
the trigonometric ratios
Sine
Use
Sine
cosine
and
tangent
the trigonometric ratio
sine, cosine, and
Cosine
to solve for missing sides in Engaging Mathematics
Find XZ .
 sine
tangent to solve problems.
Tangent
right triangles
p.127 (50.pdf)
 cosine
Answers: XZ = 14.03 in
Inverse
pp. 225-226 (89.pdf)
Readiness Standard
 tangent
Angle
of
Use
the
inverse
sine
cosine
p. 229 (90.pdf)
to solve problems.
depression
and tangent to find angle
Angle of elevation
measures
Identify the relationships of
trig functions.
Misconceptions:
 The student may confuse the different trigonometric ratios.
 The student may not be able to identify which legs are opposite or the
adjacent side to an angle.
 The student may confuse the hypotenuse in a right triangle with the leg
adjacent to a given angle.
 The student may not have the calculator set in the degree mode when
solving problems with trigonometric ratios.
2016-2017
Page 2
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
G.9(B) apply the
relationships in special
right triangles 30°-60°90° and 45°-45°-90° and
the Pythagorean
theorem, including
Pythagorean triples, to
solve problems.
Readiness Standard
Apply the relationships in
special right triangles
 30°-60°-90°
 45°-45°-90°
to solve problems.
Designated Grading Period: 5th
Days to teach: 16
Assessment
Vocabulary
Instructional
Strategies
Within a square section of
land, a landscaper will build
a path, as represented by the
shaded section in the diagram
below.
Right triangle
Hypotenuse
Opposite
Short leg
Long leg
Pythagorean
theorem
Pythagorean triples
Isosceles triangle
Having students memorize
3,4,5 and 5,12,13 triples will
help
Apply the relationships in the
Pythagorean Theorem,
including Pythagorean triples,
to solve problems.
Which measure is closest to
the value of y?
A. 5.5 ft
B. 3.1 ft
C. 4.3 ft
D. 7.5 ft
Correct answer: A
Released EOC 2013
Q # 41
2016-2017
Resources/
Weblinks
Big Ideas Geometry
9.4, 9.5
www.khanacademy.org
Remind students that the
properties of special right
triangles can be derived
from the Pythagorean
theorem
Engaging Mathematics
p.135 (53.pdf)
Supporting STAAR
Special Right Triangles
(Geom4.pdf)
Misconceptions:
 The student may not correctly identify the short leg, long leg, and
hypotenuse when solving problems involving a 30°-60°-90° triangle.

The student may multiply or divide by 2 instead of the
solving problems involving a 45°-45°-90° triangle.
2 when
Page 3
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
Designated Grading Period: 5th
Days to teach: 16
Assessment
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.(10) Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two- and threedimensional figures.
G.10(A) identify the
Identify the shapes of twoIdentify the 2-dimensional
Polyhedron
Cross-sections: Imagine
Big Ideas Geometry
shapes of twodimensional cross sections of
shapes of the cross-sections:
Face
“cutting” figures—what do
12.1
dimensional crossEdge
you see? Use tangible
 Prisms
sections of prisms,
Vertex
objects if possible. Point out www.khanacademy.org
 Pyramids
pyramids, cylinders,
Cross-section
differences if cut is parallel
 Cylinders
cones, and spheres and
Solid
of
revolution
or perpendicular to the base.
 Cones
identify threeAxis
of
revolution
 Spheres
Correct answer:
dimensional objects
Rotations: Model if

Parallel to base: circle
generated by rotations of
possible. Make sure students
Perpendicular to base:
two-dimensional shapes.
understand that a cone is
Identify three-dimensional
rectangle
created by rotating a
Supporting Standard
objects generated by rotations
triangle, a cylinder is
of two-dimensional shapes.
created by rotating a
rectangle, and a sphere is
created by rotating a circle.
2016-2017
Page 4
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
G.10(B) determine and
describe how changes in
the linear dimensions of a
shape affect its perimeter,
area, surface area, or
volume, including
proportional and nonproportional dimensional
change.
Readiness Standard
Determine and describe how
changes in the linear
dimensions of a shape affect its
 Perimeter
 Area
 Surface area, or
 Volume
Including
 Proportional
 Non-proportional
change.
Designated Grading Period: 5th
Days to teach: 16
Assessment
Given:
Vocabulary
Area
Perimeter
Surface area
Volume
Dimension changes
Instructional
Strategies
Resources/
Weblinks
Area changes by two
dimensions
Big Ideas Geometry
11.4, 12.2, 12.3, 12.4
Volume changes by three
dimensions
www.khanacademy.org
Engaging Mathematics
p. 195 (76.pdf)
By what factor does the area
of this figure change if the
height is tripled and the
width is halved? What is the
new area?
Supporting STAAR
Dimensional Change with
Area(Geom12.pdf)
Correct answer:
3
Area multiplied by
2
New area: 144 ft2
G.11 Two-dimensional and three-dimensional figures. The student uses the process skills in the application of formulas to determine measures of two-and threedimensional figures. The student is expected to:
G.11(A) apply the
Apply the formula for the area
Find the exact area of a
In a regular
Formula for area of a
Big Ideas Geometry
formula for the area of
of regular polygons to solve
regular hexagon with side
polygon:
triangle can be used to
11.3
regular polygons to
problems using appropriate
length 10 cm.
Center
develop formula for area of
solve problems using
units of measure.
Radius
a regular polygon.
www.khanacademy.org
appropriate units of
Correct answer:
Apothem
measure.
Perimeter
Engaging Mathematics
150√3 cm2
pp. 187-188 (73.pdf)
Supporting Standard
2016-2017
Page 5
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
G.11(B) determine the
area of composite twodimensional figures
comprised of a
combination of triangles,
parallelograms,
trapezoids, kites, regular
polygons, or sectors of
circles to solve problems
using appropriate units of
measure.
Readiness Standard
Find area of composite figures.
Designated Grading Period: 5th
Days to teach: 16
Assessment
Vocabulary
Instructional
Strategies
Find the area of the following
figure:
Areas of triangles,
parallelograms,
trapezoids, kites,
regular polygons,
& sectors of circles
Total area can be found by
adding smaller areas or by
subtracting areas, depending
on the shape.
Resources/
Weblinks
Big Ideas Geometry
11.3
www.khanacademy.org
Supporting STAAR
Areas of Composite
Figures (Geom7.pdf)
Correct answer:
40 in2
2016-2017
Page 6
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
G.11(C) apply the
formulas for the total and
lateral surface area of
three-dimensional figures,
including prisms,
pyramids, cones,
cylinders, spheres, and
composite figures, to
solve problems using
appropriate units of
measure.
Readiness Standard
Apply the formulas for the
 Total
 Lateral
surface area of threedimensional figures including
 Prisms
 Pyramids
 Cones
 Cylinders
To solve problems using
appropriate units of measure.
Designated Grading Period: 5th
Days to teach: 16
Assessment
The main entrance to the
Louve art museum is shaped
like a pyramid. The pyramid
is 71 feet tall and has a slant
height of approximately 91
feet. Each side of the square
base measures 115 feet.
Which of the following is
closest to the lateral surface
area of the pyramid?
A. 20,930 ft2
B. 16,330 ft2
C. 10,465 ft2
D. 34,155 ft2
Correct answer: A
Released EOC 2013
Q#37
2016-2017
Vocabulary
Lateral area
Total surface area
Prism
Pyramid
Cone
Cylinder
Composite figure
Instructional
Strategies
Point out difference in
lateral area and total surface
area: SA = LA + area of
base(s)
Resources/
Weblinks
Big Ideas Geometry
12.2, 12.3
www.khanacademy.org
In composite figures, total
surface area is almost never
the sum of the surface areas
of the smaller figures.
Misconceptions:
 The student may incorrectly convert a unit of measure when determining the
area of a two dimensional figure.
 The students may confuse the radius and diameter in a circle.
 The students may not understand the importance of the units.
 The students may confuse the slant height of a cone with the height of the
cone.
 The students may not understand the “B” in the formula represents the area of
the base of the shape.
 The students may use the formula for the circumference of a circle instead of
the formula for the area of a circle when calculating the area of the base of a
cylinder or cone.
 The students may not understand the “B” in the formula represents the area of
the base of the shape.
 The students may not understand the “P” in the formula represents the
perimeter/circumference of the base of the shape.
 The students may use the formula for the circumference of a circle instead of
the formula for the area when calculating the area of the base of a cylinder.
Page 7
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
G.11(D) apply the
formulas for the volume
of three-dimensional
figures, including prisms,
pyramids, cones,
cylinders, spheres, and
composite figures, to
solve problems using
appropriate units of
measure.
Readiness Standard
Apply the formulas for the
volume of three-dimensional
figures including
 Prisms
 Cylinders
 Composite figures
to solve problems using
appropriate units of measure.
Designated Grading Period: 5th
Days to teach: 16
Assessment
Find the volume of a cylinder
with base circumference 18π
cm and height 10cm.
Correct answer:
V=810π cm3
Vocabulary
Volume
Prism
Cylinder
Composite figure
Instructional
Strategies
Some volumes of composite
figures are found by adding
volumes, but others are
found by subtracting
volumes.
Resources/
Weblinks
Big Ideas Geometry
12.4
www.khanacademy.org
Misconceptions:
 The students may confuse the slant height of a cone with the height of the
cone.
 The students may not understand the “B” in the formula represents the
area of the base of the shape.
 The students may use the formula for the circumference of a circle instead
of the formula for the area of a circle when calculating the are of the base
of a cylinder or cone.
2016-2017
Page 8
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
Designated Grading Period: 5th
Days to teach: 16
Assessment
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.(12) Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to:
G.12(A) apply theorems
Identify tangents, secants,
Circle
Have students create a
Big Ideas Geometry
Find
about circles including
chords, diameters, radii,
Center
poster to illustrate key
10.5, 10.6
relationships among
inscribed angles, central angles,
Radius
vocab terms
angles, radii, chords,
and arc measure.
Chord
www.khanacademy.org
tangents, and secants, to
Diameter
Patty paper can be used to
solve non-contextual
Investigate properties of
Secant
how key theorems then
problems.
tangent intersections, bisecting
Tangent
Have students then sketch a
Supporting STARR
arcs and chords, secant -secant
Point of tangency
picture of each theorem to
Achievement Book
Supporting Standard
Tangent circles
keep in their notebooks
(Geom1.pdf)
Answer: 78°
Concentric circles
Common tangents
Engaging Mathematics
If AC = 17 and CD = 8, then
Central angles
p. 281 (110.pdf)
find AB.
Minor arc
p. 283 (111.pdf)
Major arc
pp. 301-302 (119.pdf)
Adjacent arcs
Inscribed angle
Intercepted arc
Circumscribed
circle
Answer: AB= 30
External segment
2016-2017
Page 9
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
G.12 (B) apply the
proportional relationship
between the measure
of an arc length of a circle
and the circumference of
the circle to
solve problems.
Supporting Standard
G.12 (C) apply the
proportional relationship
between the measure of
the area of a sector of a
circle and the area of the
circle to solve problems.
Supporting Standard
Find circumference, arc length
Designated Grading Period: 5th
Days to teach: 16
Assessment
Find the length of arc IJ.
Vocabulary
Circumference
Radius
Arc length
Central angle
Measure of an arc
Instructional
Strategies
Arc length is a fractional
part of the circumference.
Resources/
Weblinks
Big Ideas Geometry
11.1
www.khanacademy.org
Engaging Mathematics
p. 285 (112.pdf)
Find area of a circle, sector area
Correct answer:
6.25 π yd2
Jonathan’s job at the golf cou
rse consists of making sure
that the sprinklers
are properly adjusted. The
sprinklers must turn through
an angle of 320 degrees and b
e able to spray water from
0 to
70 feet from the
sprinkler. What is
the total area each sprinkler
must be able to water?
Round your answer to the
nearest square foot?
Radius
Area of a circle
Central angle
Area of a sector
Sector area is a fractional
part of the area of the circle.
Big Ideas Geometry
11.2
www.khanacademy.org
Engaging Mathematics
p. 297 (117.pdf)
Correct answer:
13,683 sq. feet
2016-2017
Page 10
Mathematics
Course: Pre-AP Geometry
Unit 9: Right Triangles and Trigonometry (cont)
Unit 10: Circles
Unit 11: Circumference and Area
Unit 12: Surface Area &Volume (begin)
TEKS
Guiding Questions/
Specificity
G.12 (D) describe radian
measure of an angle as the
ratio of the length of an
arc intercepted by a
central angle and the
radius of the circle
Supporting Standard
G.12 (E) show that the
equation of a circle with
center at the origin and
radius r is x2 + y2 = r2 and
determine the equation
for the graph of a circle
with radius r and center
(h, k), (x - h)2 + (y - k)2 =
r2.
Supporting Standard
2016-2017
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
𝑟𝑎𝑑𝑖𝑎𝑛
=
𝑚𝑒𝑎𝑠𝑢𝑟𝑒
𝑟𝑎𝑑𝑖𝑢𝑠
Show that the equation of a
circle with center at the origin
and radius r is x2 + y2 = r2 .
Designated Grading Period: 5th
Days to teach: 16
Assessment
Vocabulary
Instructional
Strategies
Find the radian measure of
the central angle of a circle of
radius 4 cm that intercepts an
arc of length 16 cm.
Degree measure of
an angle
Radian measure of
an angle
Convert between degrees
and radians; use conversion
and arc length formula.
Correct answer:
4 radians
Write the equation of and
graph a circle with center (-4,
3) with radius 6.
Equation of a circle
Center of a circle
Radius
When h and/or k is negative,
“-“ changes to “+” in
equation
Resources/
Weblinks
Big Ideas Geometry
11.1
www.khanacademy.org
Big Ideas Geometry
10.7
www.khanacademy.org
Determine that the equation for
the graph of a circle with radius
r and center (h, k),
(x - h)2 + (y - k)2 = r2.
Correct answer:
(x + 4)2 + (y - 3)2 = 36
Supporting STAAR
Extending the
Pythagorean Theorem
(Geom8.pdf)
Page 11