Download UNIT 1 Congruence, Proof, and Construction

UNIT 1
Congruence, Proof, and Construction
Total Number of Days: 26 days
ESSENTIAL QUESTIONS
1.
2.
How do you identify corresponding parts of
congruent triangles?
How can you make a conjecture and prove
that triangles are congruent?
How can you describe the attributes of a
segment or angle?
3.
Grade/Course: __Geometry 10th grade__
ENDURING UNDERSTANDINGS
4.
If two triangles are congruent, then every pair of their corresponding parts is also congruent.
5.
Given information, definitions, properties, postulates, and previously proven theorems can be
used as reasons in a proof.
6.
Number operations can be used to find and compare the lengths of segments.
7.
The Ruler and Segment Addition Postulates can be used in reasoning about lengths.
8.
The Protractor and Angle Addition Postulates can be used in reasoning about angle measures.
RESOURCES
STANDARDS
CONTENT
PACE
SKILLS
OTHER
(CCSS/MP)
Pearson
LEARNING ACTIVITIES and
ASSESSMENTS
(e.g., tech)
Square numbers when
calculating areas of certain
Getting Ready figures.
for Geometry:
2 days
Example:
Basic Skills
1) The number you get when
(Squaring
you multiply an integer by
numbers,
itself.
Evaluating
1.
8.EE.2
Students recognize perfect
Pearson
squares and cubes,
understanding that non-perfect Chapter #1
squares and non-perfect cubes Get Ready
are
Irrational.
Interactive website for
exponent:
Website related to step to solve
http://www.mathsisfun equations:
.com/exponent.html
www.svmimac.org/images/MNM.
052913.stsec.pdf
Interactive practice
1
Expressions,
Finding
Absolute Value,
Solving
4 × 4 = 16, so 16 is a square
Equations)
number.
2) 32 = 9
3) 42 = 16
Lesson Check:
8.EE.7
games:
Students solve one-variable
Text book
equations including those with page #1
the variables being on both sides
of the equal sign. Students
recognize that the solution to the
equation is the value(s) of the
variable, which make a true
equality when substituted back
into the equation. Equations shall
include rational numbers,
distributive property and
combining like terms.
Text book Get Ready page 1
classroom.jcschools.net/basic/mathexpon.html
Video:
www.mathplayground.c
om/howto_algebraeq1.
MP.1
Evaluate expressions by
substituting given values.
MP.2
-Example:
Evaluate each using the
values given:
p
Basic skills
Use
30mn. Review:
HSPA Ex
m; use m
, and p
Apply Pythagorean Theorem G.SRT.8 Use trigonometric ratios Pearson
and the Pythagorean Theorem
in real life problems:
to solve right triangles in applied Chapter #1
Example:
Get Ready
Web link for
Pythagorean Theorem
and the distance
Pythagorean Theorem Power
point presentation:
www.jamestownpublicschools.o
2
PREP/PARCC/S
problems.
AT
Michelle was fishing in her
canoe at point A in the lake
depicted above. After trying
to fish there, she decided to
paddle due east at a steady
speed of 10 miles per hour.
As she paddled, a wind
blowing due south at 5 miles
per hour caused a change in
her direction. What is the
speed of her canoe,
measured to the nearest
tenth of a mile per hour,
which has a velocity
represented by vector AC?
formula.
Text book
page #1
www.mathscore.com/m
ath/practice/Pythagore
an%20Theorem
rg/highschool/faculty/.../pythag
thm.ppt
Kuta software for
worksheets
www.kutasoftware.co
m
Michelle
1.
Nets and
Drawing for
1 day Visualizing
Geometry
Construct nets and drawings G.CO.1
of three- dimensional figures.
Know precise definitions of
Example: refer to link under angle, circle, perpendicular line,
resources on discovering 3-D parallel line, and line segment,
shapes.
based on the undefined notions
of point, line, distance along a
Example:
line, and distance around a
Find the surface area of this circular arc.
Pearson
Chapter 1
Text book
page #4-9
Interactive Animated
polyhedron models:
1.
Basic: problems 1-4 Exs
www.mathsisfun.com/g 6-19 all, 20-26 even, 27, 28-36
even,
eometry/polyhedron43-51
3-Dimentional shapes
2.
Average: problems 1-4
videos and worksheets
Exs. 7-19 odd, 20-38, 43-51
3
box below
www.onlinemathlearnin 3.
Advanced: Problems 1-4
g.com/3d-shapesExs. 7-19 odd, 20-51
nets
MP.3
MP.7
Web link for 3D shapes:
http://www.xtec.cat/mo
nografics/cirel/pla_le/nil
e/mrosa_garcia/worksh
eets.pdf
Basic
Review:
skills Identify congruent figures
and their corresponding
parts.
G.SRT.5
Use congruence and similarity
HSPA
criteria for triangles to solve
PREP/PARCC/SA Example:
problems and to prove
relationships in geometric
T
A design follows this pattern:
figures.
an equilateral triangle is
divided into 4 congruent
30mn
triangles as shown below in
Stage 1. Then, the top triangle
is divided into 4 congruent
triangles and the pattern
repeats for each stage. In
Stage 2, what is the ratio of
the area of the larger shaded
triangle to the area of the
Pearson
Chapter 4-1
Web link for congruent Standardized Test Prep
triangles:
(SAT/HSPA)
www.mathopenref.com Text book page 224 Q. 50-53
/congruenttriangles.ht
Text book
page #4-9
4
smaller shaded triangle?
1.
Points, Lines,
Planes
- To understand basic terms
and postulates of Geometry
Example:
1 day
G.CO.1
Pearson
Know precise definitions of
Chapter 1
angle, circle, perpendicular line,
parallel line, and line segment,
based on the undefined notions Text book
of point, line, distance along a page #10-19
line, and distance around a
circular arc.
MP.3
HSPA
30mn PREP/PARCC/SAT
To identify parallel and
perpendicular lines.
G-C.O.1
3.
Pearson
Chapter 1
Ex. How would you determine
whether two lines are parallel
See below
1.
Basic: problems 1-2 Exs
8-14 all, 65-80.
Problems 3-4
Exs. 15-26 all, 28-46 even, 51,
54-58 even
http://coachmetz.files.w
ordpress.com/2012/04/
geometry_point_lines_a
nd_planes_worksheet_a 2.
Average: problems 1-2
Exs. 9-13 odd, 65-80. Problems
.pdf
3-4
Exs. 15-25 odd, 27-58
MP.6
Line
Basic skills Review:
Web link for points,
lines and planes:
Advanced: Problems 1-2
Exs.9-13 odd, 65-80. Problems
3-4
Exs. 15-25 odd, 27-64
Web link for parallel
Standardized Test Prep
and perpendicular lines: (SAT/HSPA)
www.clackamasmiddlec
ollege.org/.../Parallel+a
Text book page 10 Q. 43-45
5
or
perpendicular?
1.
Measuring
Segments
1 day
Determine and compare
length of segments.
Example:
G.CO.1
Text book
page #10
nd+Perpendicular+lines.
pdf
Pearson
Web link for segments
and their measures:
Know precise definitions of
Chapter 1
angle, circle, perpendicular line,
parallel line, and line segment,
based on the undefined notions Text book
of point, line, distance along a page #20-27
line, and distance around a
circular arc.
www.kutasoftware.com
/FreeWorksheets/Geo
Worksheets/2Line%20Seg
4.
Basic: problems 1-4 Exs
8-22 all, 24-34 even, 35, 3739, 44-56
5.
Average: problems 1-4
Exs. 9-21 odd, 23-41, 44-56
6.
Advanced: Problems 1-4
Exs. 9-21 odd, 23-56
G.GPE.6
Find the point on a directed line
segment between two given
points that partitions the
segment in a given ratio.
6
MP.1, MP.3
Basic skills
Review:
30
mn
To find translation image of G-CO.6
Pearson
figures:
Use geometric descriptions of Chapter 9-1
Example:
HSPA
rigid motions to transform
PREP/PARCC/S
figures and to predict the effect
Consider parallelogram ABCD
of a given rigid motion on a
AT
Text book
with coordinates A(2,-2),
B(4,4), C(12,4) and D(10,-2). given figure; given two figures, page #552
use the definition of congruence
Perform the following
in terms of rigid motions to
transformations. Make
predictions about how the
decide if they are congruent.
lengths, perimeter, area and
angle measure will change
under each transformation.
Weblink for practice
with translation in
coordinate geometry.
Standardized Test Prep
(SAT/HSPA)
Text book page 552
www.mathsisfun.com/g Q. 36-39
eometry/translation.ht
Weblink for Video on
translation:
www.brightstorm.com/math/geo
metry/transformations/translatio
ns
a. A reflection over the x-axis.
b. A rotation of 270about the
origin. c. A dilation of scale
factor 3 about the origin. d. A
translation to the right 5 and
down 3.
Verify your predictions.
Compare and contrast which
transformations preserved the
size and/or shape with those
that did not preserve size
and/or shape. Generalize, how
could you determine if a
7
transformation would maintain
congruency from the pre-image
to the image?
1.4 Measuring Determine and compare the G.CO.1
measures of angles.
Angles
Know precise definitions of
Identify special angle pairs angle, circle, perpendicular line,
and use their relationships to parallel line, and line segment,
1-5 Exploring find angle measures.
based on the undefined notions
Angle Pairs
of point, line, distance along a
Example:
line, and distance around a
circular arc.
Web link for angles and 1.
Basic: (1.4) problems 1-2
Exs 6-17 all, Exs. 6-17 all, 41their measures:
Chapter 1
49.
http://aggiejots.tripod.c
om/sitebuildercontent/s Problems 3-4 Ex. 18-23 all, 24itebuilderfiles/geo_0106 28 even, 29, 31-34, 41-49
Text book
page # 28-47 _ans.pdf
(1.5) problems 1-2 Exs. 7-23
all, 48-59. Problems 3-4
Pearson
Exs. 7-26 all, 28-30 even,
31,34-38 even, 39-40
G.CO.12
1 day
Make formal geometric
constructions with a variety of
tools and methods (compass and
straightedge, string, reflective
devices, paper folding, dynamic
geometric software, etc.).
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
2.
Average:
1) Section (1.4) problems 1-2
Exs. 7-17 odd, 41-49.
3.
Problems 3-4 Exs. 19-23
odd, 24-32, 41-49. (1.5)
Problems 1-2 Exs. 7-23 odd,
48-59. Problems 3-4
Exs. 25, 27-41
4.
Advanced:
8
1) Section (1.4) Problems 1-4
Exs.7-23 odd, 24-49.
point not on the line.
2) Section (1-5) Problems 1-4
Exs. 7-25 odd, 27-59
MP.1, MP.3
Basic skills
Review:
G.SRT.8
Pearson
Use the Pythagorean
Theorem to solve real life
Use trigonometric ratios and the Chapter 5
problems:
HSPA
Pythagorean Theorem to solve
Example:
PREP/PARCC/S
right triangles in applied
AT
A 16-ft ladder leans against problems.
Text book
30mn
1.6
Basic
1 day Constructions
a building. To the nearest
foot, how far is the base of
the ladder from the
building? Sketch the
diagram.
page # 291
Perform basic constructions G.CO.12
using a straightedge and
Make formal geometric
compass.
constructions with a variety of
Example:
tools and methods (compass
and straightedge, string,
Construct a circle
reflective devices, paper folding,
circumscribed about triangle
Pearson
Kuta software for
worksheet:
Standardized Test Prep
(SAT/HSPA)
www.kutasoftware.com Text book page 498 Q. 55-58
/FreeWorksheets/PreAlg
Worksheets/Pythagorea
n
Interactive game link for
Pythagorean Theorem:
www.mathplay.com/PythagoreanTheorem-Game
Chapter 1
Compass and
5.
Basic:
straightedge
construction worksheets 1) problems 1-2
available on this site:
Exs 7-12 all, 20, 39-47.
www.mathopenref.com
Text book
page # 48-56 /worksheetlist
2) Problems 3-4
9
ABC
dynamic geometric software,
etc.). 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.
Exs. 13-16 all, 18, 19, 22,
24, 25, 26-30 even, 36-38
6.
Average:
1) problems 1-2
Exs. 7-11 odd, 20, 39-47.
2) Problems 3-4
Exs. 13, 15, 17-19, 21-32,
36-38
MP.1, MP.3, MP.5, MP.7
7.
Advanced:
1) Problems 1-2
Exs. 7-11 odd, 20,
39-47
2) Problems 3-4
Exs. 13, 15, 17-19,
21-38
Basic skills
Review:
Use different scale to
solve real life
applications
HSPA
Example:
30mn PREP/PARCC/S
AT
Jan is building a scale
model of a house. If the
actual house is 86 feet
wide and 172 feet long,
G.GMD.3
Pearson
Use volume formulas for
Chapter 10-1
cylinders, pyramids, cones, and
spheres to solve problems.
Text book
page # 622
Web link for area and
parallelogram:
Standardized Test Prep
(SAT/HSPA)
Text book page 622 Q. 47-49
http://www.mathgoodie Kutasoftware for worksheet:
s.com/lessons/vol1/area
www.kutasoftware.com/FreeW
_parallelogram.html
orksheets/PreAlgWorksheets/
Area
10
what will be the length in
inches of the scale model if
it is 18 inches wide?
G.GPE.4
1.7 Midpoint Find the midpoint of a
and Distance in segment.
the Coordinate
Plane
1 day
Pearson
Use coordinates to prove simple Chapter 1
geometric theorems
algebraically. For example,
prove or disprove that a figure Text book
defined
page #49-66
Find the distance between
two points in the coordinate
by four given points in the
plane.
coordinate plane is a rectangle;
Example:
prove or disprove that the point
(1, 3) lies on the circle centered
The distance from the floor to
at the origin and containing the
the ceiling of a rectangular
point (0, 2).
room is 8 ft. The diagonals of
two adjacent walls are 17 ft
and
ft, respectively.
G.PE.7
How long is a diagonal of the
Use coordinates to compute
floor?
perimeters of polygons and
areas of triangles and
rectangles, e.g., using the
distance formula.
MP.1, MP.3
Web link for
8.
Pythagorean theorem
and the distance
formula:
Basic:
1) problems 1-2
Exs. 6-21 all, 62-67
http://users.manchester 2) Problems 3-4
.edu/Student/slmiller02
Exs. 22-35all,
/ProfWeb/PythagoreanT
36-44 even, 45-47 all, 48-56
heoremNotes.pdf
even
9.
Average:
1) problems 1-2
Exs. 7-21 odd, 62-72
2) problems 3-4
Exs. 23-35 odd, 36-57
10.
Advanced:
1) Problems 1-2
Exs. 7-21 odd, 62-72
2) Problems 3-4
11
Exs. 23-35 odd, 36-61
Basic skills
Review:
Apply trigonometric ratios.
Example:
G.SRT.8
Pearson
Use trigonometric ratios and the Chapter 8-1
HSPA
Pythagorean Theorem to solve
Joshua is flying his kite at the
PREP/PARCC/S
right triangles in applied
end of a100ft. string. The
AT
problems.
Text book
angle of the string to the
page # 498
ground is 50 degrees.
Find the
30
mn
height, x,
of the kite
above
the ground.
Web link for
trigonometry ratios:
11.
Standardized Test Prep
(SAT/HSPA)
http://www.kutasoftwar
Text book page 498 Q. 55-58
e.com/FreeWorksheets/
GeoWorksheets/9Trigonometric%20Ratios 12.
Web link for video
.pdf
tutorial on trigonometric
ratio
www.youtube.com/watch?
12
2.1
Patterns and
inductive
reasoning
1 day
Use inductive reasoning to
make conjectures.
Weblink for an
13.
Basic:
application of inductive
Prove theorems about lines and Chapter 2-1 reasoning:
1) problems 1-3
Example:
angles. Theorems include:
Exs. 6-30
vertical angles are congruent;
www.rohan.sdsu.edu/~i
Show the conjecture is false
2) Problems 4-5
when a transversal crosses
tuba/math303s08/mathi
Text book
by finding a counterexample.
parallel lines, alternate interior page # 82-88
a
i
f
Exs. 31-40, 50, 53,54, 59Conjecture: The sum of two angles are congruent and
66
corresponding angles are
numbers is always greater
14.
Average:
congruent; points on a
than the larger of the two
perpendicular bisector of a line
numbers.
1) problems 1-3
segment are exactly those
qui i an fr
g n’
Exs. 7-29, 38-49
endpoints.
2) problems 4-5
Prepare for: G.CO.9
Pearson
Exs. 31-37 odd,
50-55, 59-66
15.
Advanced:
1) Problems 1-3
Exs. 7-29 odd,
38-49
2) Problems 4-5
Exs. 31-37 odd,
50-66
13
Basic skills
Review:
Apply the surface area and
the volume formulas:
Example:
30
mn
HSPA
PREP/PARCC/S
Find the approximate surface
AT
area of this can to the nearest
square inch. The diameter of
the top is about 6 inches and
the height of the cylinder is 8
inches.
G.GMD.3
Pearson
Use volume formulas for
Chapter 11
cylinders, pyramids, cones, and
spheres to solve problems.
Text book
page #695
Web link for surface
area and volume:
16.
www.mathatube.com/c
ylinder-volume-surfacearea-worksheets
Standardized Test Prep
(SAT/HSPA)
Text book page 695 Q. 51-55
17.
Web link for video
tutorial on surface area
and volume:
www.khanacademy.org/.../c
ylinder-volume-and-surfacearea
14
2.2
Conditional
statements
1 day
Compose and distinguish
Prepares for:
between the converse,
inverse, and contrapositive G.CO.9 Prove theorems about
of a conditional statement. lines and angles. Theorems
include: vertical angles are
Example:
congruent; when a transversal
crosses parallel lines, alternate
Your classmate claims that
interior angles are congruent
the conditional and
and corresponding angles are
contrapositive of the
congruent; points on a
following statement are both
perpendicular bisector of a line
true. Is he correct? Explain
segment are exactly those
2
qui i an fr
g n’
If X = 2, then X = 4
endpoints.
1.
Can you find a
counterexample of the
conditional?
2.
Do you need to find a
counter example of the
contrapositive to know its
truth table?
Pearson
Chapter 2-2
Web link for conditional 3. Basic:
statement:
1) Problems 1-4
http://wwwregensprep.
Exs. 5-24, 28-29, 35, 37
org/Regents/n
Text book
page #89-96
39-40, 47-58
Kuta software for
worksheet:
www.kutasoftware.co
m
4. Average:
1) problems 1-4
Exs. 5-23, 25-42, 47-58
5. Advanced:
you tube video tutorial:
www.youtube.com/wat
ch?v=undSZSratIA
1) Problems 1-4
Exs. 5-23 odd, 25-58
Math power point notes:
Google search:
ac r
nric k va u … C n i
titionalStatement/LECTURE2-1.ppt
www a c
rg … G
ryPP
Ts/2.1%20Conditional%20Statement
www a
a c
… C a r%
20Powerpoints/2.2%20Intro%2
Math skills practice:
15
Intranet.asfa.k.12 a u … Law %
f%20Logic
16
Basic skills
Review:
HSPA
PREP/PARCC/S
AT
30
mn
To apply the Pythagorean
G.SRT.8 Use trigonometric ratios Pearson
Theorem and the perimeter and the Pythagorean Theorem
to solve right triangles in applied Chapter 2-2
formula:
problems.
Example:
The backyard behind Mr.
J n n’
u i a
rectangle. A sidewalk from
one corner of the backyard to
the opposite corner is 76 feet
long. Both the backyard and
the house are 40 feet wide.
What is the approximate
perimeter of the backyard?
Kuta software for
worksheet:
6.
Standardized Test Prep
(SAT/HSPA)
Text book page 95 Q. 47-50
7.
Interactive game link for
Pythagorean Theorem:
www.kutasoftware.co
m
Text book
page #95
www.mathplay.com/PythagoreanTheorem-Game.html
Sketch the figure.
17
Illustrate bi-conditionals and Prepares for
Pearson
recognize good definitions.
Bi-conditionals
G.CO.10
Chapter 2-3
Example:
and Definitions
Prove theorems about triangles.
What are the two conditional Theorems include: measures of
statements that form this bi- interior angles of a triangle sum Text book
conditional?
to 180°; base angles of isosceles page #99-105
triangles are congruent; the
A ray is an angle bisector if
segment joining midpoints of
and only if it divides an angle
two sides of a triangle is parallel
into two congruent angles.
to the third side and half the
length; the medians of a triangle
meet at a point.
2.3
1 day
Web link for bi8.
conditional statement:
http://www.mathgoodie
s.com/lesson/
Basic:
1) Problems 1-3
Exs. 7-30, 33, 35-36,
43, 45, 49-57
Kuta software for
worksheet:
www.kutasoftware.co
m
9.
Average:
1) problems 1-4
Exs. 7-30, 33, 35-36,
43, 45, 49-57
you tube video tutorial:
www.youtube.com/wat
ch?v=12CeL-hFky8
10.
Advanced:
1) Problems 1-4
Exs. 7-29 odd, 30-57
Math power point notes:
Google search:
teachers.henric k va u … C n
dititionalStatement/LECTURE21.ppt
www a c
rg … G
f
c
fu i n u
/get_group_file.phtml?...
u
ry…
www.ohio.edu/people/melkonia/
math306/slides/logic2.ppt
18
Math skills practice:
In ran a fa k
0of%20Logic
a u … Law %
PDF worksheet link:
http://mycoursecan.com/Files/Sub
jects/Ge
19
Basic skills
Review:
Apply reflections:
Example:
G.CO.1
Pearson
Use the undefined notion of a Chapter 2-3
HSPA
point, line, distance, along a line
PREP/PARCC/S
and distance around a circular
arc to develop definitions for
AT
is the image when
Text book
angles,
circles,
parallel
lines,
point F is reflected over
page #104
perpendicular lines and line
the line
and
segments.
then over the line
The location of
30
mn
is
Which of
the following is the
location of point F ?
Web link on reflection: 11.
www.regentspre.org/Re
gents/math/geometry/G
T1/reflect.htm
Standardized Test Prep
(SAT/HSPA)
Text book page 104 Q. 49-51
12.
Interactive game link for
Pythagorean Theorem:
www.mangahigh.com/en_us
/games/translar
a.
b.
c.
d.
20
2-4
Deductive
reasoning
1 day
Prepares for:
Apply the law of detachment G.CO.11
and the law of syllogism:
Prove theorems about
Example:
parallelograms. Theorems
include: opposite sides are
What can you conclude from
congruent, opposite angles are
the given true statement?
congruent, the diagonals of a
If a u n g an “A” n a parallelogram bisect each other,
final exam, then the student and conversely, rectangles are
parallelograms with congruent
will pass the course.
diagonals.
Pearson
Chapter 2-4
Text book
page # 106112
Web links for
bi-conditional
statements:
rewww.khanacademy.or
g/.../geometry.../cageometry--deductive-
13.
Basic:
1) Problems 1-3
Exs. 6-21, 26, 28, 30,
33-39
Average:
www.sparknotes.com/... 14.
/geometry3/inductivean
1) problems 1-3
ddeductivereasoning/se
ct..
Exs. 7-17 odd, 18-30,
Kuta software for
worksheet:
33-39
15.
Advanced:
www.gobookee.net/ded
uctive-reasoning-kuta/
1) Problems 1-3
www.mybookezz.org/ge 17 odd, 18-39
Exs. 7-
ometric-mean-kutasoftware-1344/
you tube video tutorial: Math power point notes:
www.khanacademy.org/ Google search:
.../geometry.../cawww.cecs.csulb.edu/~mopkins/cecs100/D
geometry--deductive-... eductInduct.pptx
www.youtube.com/wat
ch?v=GluohfOedQE
www.taosschools.org/.../GeometryPPTs/2
.3Deductive%20Reasoning.ppt
21
Math skills practice:
www.csun.edu/~kme52026/Chapter4.pdf
www.brighthubeducation.com ›
Lessons: Grades 9-12
› Math
PDF worksheet link:
www.frapanthers.com/.../Geometry(
H)/worksheets/WorksheetDeductiv
e
www.matsuk12.us/cms/lib/AK010009
53/Centricity/.../geo2_1WS.pdf
22
Basic skills
Review:
G.CO.2-5
Develop and perform rigid
HSPA
transformations that include
Triangle ABC is shown in the
PREP/PARCC/S
reflections, rotations,
coordinate plane below. Draw translations and dilations using
AT
the result of the
geometric software, graph
transformation when triangle paper, tracing paper, and
ABC is translated 6 units to
geometric tools and compare
the right and then rotated
them to non-rigid
clockwise about the
transformations.
origin.
30
Apply transformations:
Pearson
Section
2-4
More HSPA
PREP Text
book page
#112
Q. 33-34
Kuta software for
16.
Standardized Test Prep
worksheet
(SAT/HSPA)
www.kutasoftware.com
Text book page 112 Q. 33,
/FreeWorksheets/GeoW
34
orksheets/12All%20Tran
17.
Interactive game link for
Pythagorean Theorem:
www.kidsmathgamesonline.
com/geometry/transformati
on.html
mn
23
2-5
Prepares for:
Connect reasoning in G.CO.9
algebra and
Prove theorems about lines and
geometry.
angles. Theorems include:
Example:
vertical angles are congruent;
when a transversal crosses
What is the name of the
parallel lines, alternate interior
property of equality or
angles are congruent and
congruence that justify going
corresponding angles are
from the first statement to
congruent; points on a
the second statement?
perpendicular bisector of a line
segment are exactly those
qui i an fr
g n’
1.
2x + 9 =19
endpoints.
Reasoning in
Algebra and
Geometry
1 day
2.
and
so
3.
Pearson
Chapter 2-5
Text book
page # 113119
Web link for reasoning 4.
Basic:
in algebra and
1) Problems 1-3
geometry:
Exs. 5-17, 20, 22, 23
www.mathplayground.c
om/games.html
29-41
5.
www.xpmath.com
Average:
1) problems 1-3
Exs. 5-13 odd, 14-24,
Kuta software for
worksheet:
www.gobookee.net/geo
metry-review-5-kutaanswers/
29-41
6.
Advanced:
1) Problems 1-3
Exs. 5-13 odd, 14-41
Math power point notes:
you tube video tutorial:
Google search:
www.cvsd.org/.../Geometry.../2www.youtube.com/wat
5%20Reasoning%20in%20Algebra
ch?v=xkTgnN5pOks
%20a
vimeo.com/49030863
jcs.k12.oh.us/.../Geometry/PH_G
eo_2a ning in A g ra
Math skills practice:
www.nhvweb.net/nhhs/math/ms
chuetz/files/.../Section-2-5-and-224
6.pdf
www.brighthubeducation.com ›
› Math Lessons: Grades 9-12
PDF worksheet link:
www.frapanthers.com/.../Geome
try(H)/worksheets/WorksheetDed
uctive
www.quia.com/files/quia/users/a
lamed/Geoguide/Geoguide2.5
25
Basic skills
Review:
Use reflections, rotations, and
HSPA
transformations:
PREP/PARCC/S
Example:
AT
Triangle ABC and triangle
LMN are shown in the
coordinate plane below.
G.CO.6-8
Pearson
Use rigid transformations to
determine, explain and prove
congruence of geometric
figures.
Section
2-5
More HSPA
PREP Text
book page
#119
Q. 29-33
Kuta software for
worksheet:
7.
www.kutasoftware.com
/freeige.html
Standardized Test Prep
(SAT/HSPA)
Text book page 119
Q. 29- 33
8.
Interactive game link for
Pythagorean Theorem:
www.mathplayground.com/
ShapeMods/ShapeMods.ht
ml
30
mn
Part A: Explain why triangle
ABC is congruent to triangle
LMN using one or more
reflections, rotations, and
translations.
Part B: Explain how you can
use the transformations
described in Part A to prove
triangle ABC is congruent to
triangle LMN by any of the
criteria for triangle
26
congruence (ASA, SAS, or
SSS).
27
2.6
Proving Angles Prove and apply theorems
about angles:
Congruent
Example:
Write a paragraph proof:
Given:
are supplementary.
are
supplementary.
Prove:
G.CO.9
Pearson
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; points on a
perpendicular bisector of a line
segment are exactly those
qui i an fr
g n’
endpoints.
Chapter 2-6
Text book
page # 120127
Web link for proving
9.
theorems about angles:
www.mathwarehouse.c
om/.../triangles/similartriangle-theorems.php
Basic:
1) problems 1-2
Exs. 6-12, 46-48
2) Problems 3
Exs. 13-14, 20-21, 25,
www.khanacademy.org/
26, 28
.../angles/v/angleAverage:
bisector-theorem-proo 10.
1) problems 1-2
Kuta software for
worksheet:
Exs. 7-11 odd, 36-48
2) problems 3
www.letspracticegeome
try.com/free-geometryworksheets/
Exs. 13-30
11.
3
you tube video tutorial:
www.youtube.com/wat
ch?v=gq1B3ceW4TE
2
day
Advanced:
1) Problems 1-2
Exs. 7-11 odd, 36-48
2) Problems 3
1
Exs. 13-35
www.youtube.com/wat
ch?v=G_RsPC2dKHM
Math power point notes:
Google search:
2
www.taosschools.org/.../GeometryP
PTs/4.4%20ASA%20AND%20AAS 28
www.dgelman.com/powerpoints/.../
2.6%20Proving%20Statements%20a.
.
Math skills practice:
Basic skills
Review:
HSPA
PREP/PAR
CC/SAT
G.CO.6, 7, 8
Use transformations:
Use rigid transformations to
determine, explain and prove
Quadrilateral PQRS is shown
congruence of geometric
below. Which of the following
figures.
transformations of triangle
PTS could be used to show
that
triangle PTS is congruent to
triangle QTR ?
Pearson
Section
2-6
More HSPA
PREP Text
book page
#127
Kuta software for
worksheet:
12.
ww2.d155.org/.../Geom
etry%20363%20.../Ch%2
%
% Pack
Standardized Test Prep
(SAT/HSPA)
Text book page 127
Q. 36- 39
13.
Interactive game link for
transformation
www.onlinemathlearning.co
m/transformation-ingeometry
30
mn
1) A reflection over segment QS
2) A reflection over segment PR
29
3) A reflection over line m
4) A reflection over line l
30
3.1
Lines and
Angles
Identify relationships
between figures in space.
Identify angles formed by
two lines and a transversal.
Example:
Think of each segment in the
diagram as part of a line.
Which of the lines appears to
fit the description?
a. Parallel to line AB and
contains D
1 day
b. Perpendicular to line AB
and contains D
c. Skew to line AB and
contains D
d. Name the plane(s) that
contain D and appear to be
parallel to plane ABE.
G.CO.1
Pearson
Use the undefined notion of a
point, line, distance, along a line
and distance around a circular
arc to develop definitions for
angles, circles, parallel lines,
perpendicular lines, and line
segments.
Chapter 3-1
Text book
page # 140146
Web link for properties 14.
Basic:
and parallel lines:
1) problems 1-3
www.nexuslearning.net/
Exs. 11-29 all, 30-44
.../ML%20Geometry%20
3-1%20Lines% an %
Even, 49-60
www.khanacademy.org/ 15.
Average:
.../angles/v/angle1) problems 1-3
bisector-theorem-proo
Exs. 11-23 odd, 25, 45, 49-60
Kuta software for
16.
Advanced:
worksheet:
www.kutasoftware.com 1) Problems 1-3
/.../13Exs. 11-23 odd, 25 - 60
Line%20Segment%20Co
n ruc n
f
Math power point notes:
you tube video tutorial: Google search:
www.khanacademy.org/ www.eht.k12.nj.us/~Simmonsg/lines
.../segments.../lines-- %20ppt.ppt
line-segments--and-ra..
www2.carrollk12.org/instruction/el
www.youtube.com/wat emcurric/.../line%20powerpoint.
ppt
c v
nv wI
Math skills practice:
www.mathwarehouse.com/.../triangles/si
milar-triangle-theorems.php
31
www.sanjuan.edu/webpages/john
higgins/files/81%20Practice
PDF worksheet link:
tms6thgrade.weebly.com/uploads
/8/5/1/7/8517785/practice_81
32
Basic skills
Review:
HSPA
PREP/PAR
CC/SAT
G. CO.12, 13
Use perpendicular lines
Example:
Use paper folding to
construct the perpendicular
bisector of line segment
shown below. Trace and label
the line segment JK.
Generate formal constructions
with paper folding, geometric
software and geometric tools to
include, but not limited to, the
construction of regular polygons
inscribed in a circle.
Pearson
Section
3-1
More HSPA
PREP Text
book page
#146
Kuta software for
worksheet:
17.
Standardized Test Prep
(SAT/HSPA)
Text book page 146 Q. 49- 52
www.kutasoftware.com
/FreeWorksheets/Geo
Worksheets/218.
Interactive game link for
Line%20Seg
transformation
www.sheppardsoftware.com/
mathgames/geometry/.../line
_shoot.htm
30
mn
33
3.2 properties
of parallel lines
2
days
Pearson
Prove theorems about
parallel lines.
G.CO.9
Chapter 3-2
Prove theorems about lines and
Use properties of parallel
angles. Theorems include:
lines to find angle measures. vertical angles are congruent; Text book
page # 147when a transversal crosses
Example:
parallel lines, alternate interior 155
angles are congruent and
Complete the proof of the
Consecutive Interior Angles corresponding angles are
congruent; points on a
Theorem.
perpendicular bisector of a line
GIVEN: p q
segment are exactly those
qui i an fr
g n’
PROVE: 1 and 2 are
endpoint.
supplementary.
Web link for reasoning
19.
in algebra and
geometry:
www.slideshare.net/1co
nejo/proving-lines-areparallel
web.mnstate.edu/peil/g
20.
eometry/.../6ExteriorAn
gleR.htm
Kutasoftware for
worksheet:
www.kutasoftware.com
fr ig
you tube video tutorial:
21.
www.youtube.com/wat
c v
L
G
Basic:
1) problems 1-2
Exs. 7-11, 29-39
2) Problems 3-4
Exs. 12-18, 29-39
Average:
1) problems 1-2
Exs. 7-11 odd, 29-39
2) problems 3-4
Exs. 13-17 odd,18-26
Advanced:
1) Problems 1-2
Exs. 7-11 odd, 29-39
2) Problems 3-4
Exs. 13-17 odd,18-28
Math power point notes:
Google search:
teachers.henrico.k12.va.us/.../02Perp
endicularParallel/...5ProvingLinesP
a...
34
geometryf.mths.schoolfusion.us/modules/.../get_gro
Math skills practice:
www.nexuslearning.net/.../ML%20Ge
ometry%203-3%20Parallel%20Lin
PDF worksheet link:
www.bowerpower.net/geometry/ch03/
Wksh%203.2B.pdf
35
Basic skills
Review:
HSPA
PREP/PAR
CC/SAT
12.
Prove theorems about
parallel lines.
Example:
Using the figure above and
30
the fact that line
is parallel
mn
to segment
prove that
the sum of the angle
measurements in a triangle is
G.CO.9, 10, 11
Pearson
Create proofs of theorems
involving lines, angles, triangles,
and parallelograms.* (Please
note G.CO.10 will be addressed
again in unit2 and G.CO.11 will
be addressed again in unit 4)
Section
3-2
More HSPA
PREP Text
book page
#155
Kuta software for
worksheet:
22.
Standardized Test Prep
(SAT/HSPA)
Text book page 155
Q. 29- 32
23.
Interactive game link for
transformation
www.kutasoftware.com
fr ig
www.onlinemathlearning.co
m/proving-parallellines
Use as many or as
few rows in the table as
needed.
36
3.3
Proving Lines
are Parallel
2days
G.CO.9
To determine
whether two lines
are parallel.
Prove theorems about lines and
angles. Theorems include:
vertical angles are congruent;
when a transversal crosses
Example:
parallel lines, alternate interior
In the diagram at the right, angles are congruent and
corresponding angles are
each step is parallel to the
congruent; points on a
step immediately below it
perpendicular bisector of a line
and the bottom step is
parallel to the floor. Explain segment are exactly those
g n’
why the top step is parallel to qui i an fr
endpoint.
the floor.
Pearson
Chapter 3-3
Text book
page # 156163
Web link for proving 24.
Basic:
lines are parallel
1) problems 1-2
https://dionmath.wikisp
Exs. 7-11, 18-28 even
aces.com/.../3.5+Showi
ng+Lines+are+Parallel.p
2) Problems 3-4
pt...
Exs. 12-16 all, 30-34
https://s3.amazonaws.c
Even, 35, 36,38, 47-57
om/engrademyfiles/.../03-0525.
Average:
Kutasoftware for
worksheet:
1) problems 1-2
Exs. 7-11 odd, 17-28
www.kutasoftware.com
2) problems 3-4
37
/.../3Parallel%20Lines%20an
% ran v r a
Exs. 13-15 odd, 29-41,
47-57
26.
Advanced:
you tube video tutorial:
1) Problems 1-2
www.khanacademy.org/
Exs. 7-11 odd, 17-28
.../parallel...lines/.../ide
2) Problems 3-4
ntifying-parallelExs. 13-15 odd, 28-57
Math power point notes:
Google search:
teachers.henrico.k12.va.us/.../02
PerpendicularParallel/...5Proving
LinesPa
Math skills practice:
www.regentsprep.org/Regents/
math/ALGEBRA/AC3/pracParallel
PDF worksheet link:
glencoe.mcgrawhill.com/sites/dl/free/007888484
5/634463/geohwp.pdf
38
Basic skills
Review:
In
HSPA
PREP/PAR
CC/SAT
30
mn
Proving that a
quadrilateral is a
parallelogram.
In the quadrilateral below,
and
Prove that the quadrilateral is a
parallelogram.
G.CO.9
Pearson
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; points on a
perpendicular bisector of a line
segment are exactly those
qui i an fr
g n’
endpoint
Section
3-3
More HSPA
PREP Text
book page
#163
Kuta software for
worksheet:
27.
www.kutasoftware.com
/.../3Proving%20Lines%20Par
allel f
Standardized Test Prep
(SAT/HSPA)
Text book page 163
Q. 47- 51
28.
Interactive game link for
transformation
www.mathplay.com/AnglesJeopardy/AnglesJeopardy.html
Write an informal proof.
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
___________________
39
4.4
Using
corresponding
parts of
congruent
triangles.
Apply triangle congruence
and corresponding parts of
congruent triangles.
Prove that parts of two
triangles are congruent.
Example:
1day
G.SRT.5
Pearson
Use congruence and similarity
criteria for triangles to solve
problems and to prove
relationships in geometric
figures.
Chapter 4-4
Text book
page # 244256
Web link for proving
29.
Basic:
that two parts of two
1) problems 1-2
triangles are congruent.
www c a z n c
›
Geometry Concepts and
Skills › Chapter 5
Exs. 5-8 all,
10-16 even
17, 20, 23-32
salinesports.org/mr_fre 30.
Average:
derick/GeomCS/Unit%2
1) problems 1-2
g
an
f
Kutasoftware for
worksheet:
Exs. 5, 7, 9-20,
23-32
Advanced:
www.kutasoftware.com 31.
/.../41) Problems 1-2
Congruence%20and%20
Triangles f
Exs. 5-7, 9-32
you tube video tutorial: Math power point notes:
Google search:
40
www.khanacademy.org/ www.grossmont.edu/carylee/Ma
.../congruent126/lectures/Chapter14.ppt
triangles/...triangle/.../fi
Math skills practice:
n i
www.regentsprep.org/Regents/
math/geometry/GP4/PracCongTr
i
PDF worksheet link:
www.kutasoftware.com/FreeWo
rksheets/GeoWorksheets/4Congruence
41
Basic skills
Review:
HSPA
PREP/PAR
CC/SAT
Prove vertical angles are
congruent:
G.CO.9
Pearson
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; points on a
perpendicular bisector of a line
segment are exactly those
qui i an fr
g n’
endpoint
Section
4-4
More HSPA
PREP Text
book page
#224
Kuta software for
worksheet:
32.
www.kutasoftware.com
/.../4Right%20Triangle%20Co
ngruence f
Standardized Test Prep
(SAT/HSPA)
Text book page 224 Q. 5053
33.
Interactive game link for
transformation
www.mathsisfun.com/geom
etry/trianglescongruent.html
30
mn
U
Using the figure above, prove
that vertical angles are
congruent. Use as many or as
few rows in the table as
needed.
42
4.5
Isosceles and
Equilateral
Triangles
2
days
G.CO.10
Pearson
Apply properties of isosceles Prove theorems about triangles. Chapter 4-5
and equilateral triangles.
Theorems include: measures of
interior angles of a triangle sum
Example:
to 180°; base angles of isosceles Text book
Rock Climbing
triangles are congruent; the
page # 250In one type of rock climbing, segment joining midpoints of
257
climbers tie themselves to a two sides of a triangle is parallel
rope that is supported by
to the third side and half the
anchors. The diagram shows a length; the medians of a triangle
red and a blue anchor in a
meet at a point.
horizontal slit in a rock face.
Web link for
34.
Basic:
applying properties
1) problems 1-2
of isosceles and
equilateral triangles.
Exs. 6-12 all, 37-44
www.mathworksheet.org/isosceles
-and-equilateraltriangles
2) Problems 3
Exs. 13-15,
16-24 even
www.kutasoftware.com
28-32 even
/FreeWorksheets/GeoW
35.
Average:
orksheets/4Isosceles%
1) problems 1-2
Kutasoftware for
43
worksheet:
Exs. 7-11 odd, 37-44
www.kutasoftware.com
2) problem 3
/FreeWorksheets/GeoW
Exs. 13-32
orksheets/436.
Advanced:
Isosceles%
you tube video tutorial:
1) Problems 1-3
Exs. 7-13 odd, 14-44
www.khanacademy.org/
...triangles/.../equilatera Math power point notes:
l-and-isoscelesGoogle search:
yourcharlotteschools.net/Schools
/PCHS/Dubbaneh_site/4.5a.ppt
Math skills practice:
library.thinkquest.org/20991/tex
tonly/quizzes g q
PDF worksheet link:
www.kutasoftware.com/FreeWo
rksheets/GeoWorksheets/4Isosceles%
44
Basic skills
Review:
HSPA
PREP/PAR
CC/SAT
30
mn
G.CO.11 Prove theorems about
parallelograms. Theorems
Prove that two angles or line include: opposite sides are
segments are congruent:
congruent, opposite angles are
In isosceles ABC, the vertex congruent, the diagonals of a
parallelogram bisect each other,
angle is A. What can you
and conversely, rectangles are
prove?
parallelograms with congruent
1.
AB = CB
diagonals.
2.
Pearson
Section
4-5
More HSPA
PREP Text
book page
#256
3.
4.
Kuta software for
worksheet:
5.
www.kutasoftware.com
/FreeWorksheets/GeoW
orksheets/4Isosceles%
Standardized Test Prep
(SAT/HSPA)
Text book page 256 Q. 3740
6.
Interactive game link for
transformation
www.mathsisfun.com/trian
gle
BC = AC
INSTRUCTIONAL FOCUS OF UNIT
In previous grades, students were asked to draw triangles based on given measurements. Students also have prior experience with rigid motions:
translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit,
students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. Students use triangle congruence as a
familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about lines,
angles, triangles, quadrilaterals, and other polygons. Students also apply reasoning to complete geometric constructions and explain why
constructions work.
PARCC FRAMEWORK/ASSESSMENT
7.
PARCC EXEMPLARS: www.parcconline.org (copy & paste the URL or link into search engine)
8.
Dollar Line: http://balancedassessment.concord.org/hs033.html
Example:
45
Think of a situation which could be represented by the graph below, Write a
full description of this situation (be sure to tell what each axis represents in your story.)
9.
10.
How would you determine whether two lines are parallel or perpendicular?
Consider parallelogram ABCD with coordinates A(2,-2), B(4,4), C(12,4) and D(10,-2). Perform the following transformations. Make predictions about how the
lengths, perimeter, area and angle measure will change under each transformation.
a. A reflection over the x-axis.
b. A rotation of 270about the origin.
c. A dilation of scale factor 3 about the origin.
d. A translation to the right 5 and down 3.
Verify your predictions. Compare and contrast which transformations preserved the size and/or shape with those that did not preserve size and/or shape. Generalize, how
could you determine if a transformation would maintain congruency from the pre-image to the image?
11.
Prove that any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the line.
46
12.
A carpenter is framing a wall and wants to make sure his the edges of his wall are parallel. He is using a cross-brace. What are several different ways he could
verify that the edges are parallel? Can you write a formal argument to show that these sides are parallel? Pair up with another student who created a different
argument than yours, and critique their reasoning. Did you need to modify the diagram in anyway to help your argument?
13.
Andy and Javier are designing triangular gardens for their yards. Andy and Javier want to determine if their gardens that they build will be congruent by looking
at the measures of the boards they will use for the boarders, and the angles measures of the vertices. Andy and Javier use the following combinations to build their
gardens. Will these combinations create gardens that enclose the same area? If so, how do you know?
a. Each garden has length measurements of 12ft, 32ft and 28ft.
b. Both of the gardens have angle measure of 110, 25and 45.
c. One side of the garden is 20ft another side is 30ft and the angle between those two boards is 40.
d. One side of the garden is 20ft and the angles on each side of that board are 60and 80.
e. Two sides measure 16ft and 18ft and the non-included angle of the garden measures 30.
Wiki page for Common Core Assessments:
http://maccss.ncdpi.wikispaces.net/file/view/Geometry%20Unpacking062512%20.pdf/359510147/Geometry%20Unpacking062512%20.pdf
PARCC Framework Assessment questions with Model Curriculum Website for all units:
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf
47
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGeo_Nov2012V3_FINAL.pdf
http://www.nj.gov/education/assessment/hs/hspa_mathhb.pdf
21ST CENTURY SKILLS
(4Cs & CTE Standards)
1.
Career Technical Education (CTE) Standards
1.
st
21 Century Life and Career Skills: All students will demonstrate the creative, critical thinking, collaboration, the problem-solving skills needed to function
successfully as both global citizens and workers in diverse ethnic and organizational cultures.
9.1.12.B.1: Present resources and data in a format that effectively communicates the meaning of the data and its implications
for solving problems, using multiple perspectives.
2.
Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning,
savings, investment, and charitable giving in the global economy.
9.2.12.B.3: Construct a plan to accumulate emergency “rainy day” funds.
3.
Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the process of career awareness, exploration, and
preparation in order to navigate the globally competitive work environment of the information age.
9.3.12.C.2: Characterize education and skills needed to achieve career goals, and take steps to prepare for postsecondary
options, including making course selections, preparing for and taking assessments, and participating in extra-curricular
48
activities.
4.
Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for careers in
emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees.
9.4.12.B.4: Perform math operations, such as estimating and distributing materials and supplies, to complete
classroom/workplace tasks.
Project Base Learning Activities:
http://www.achieve.org/files/CCSS-CTE-Task-FramingaHouse-FINAL.pdf
MODIFICATIONS/ACCOMMODATIONS
1.
Group activity or individual activity
2.
Review and copy notes from eno board/power point/smart board etc.
3.
Group/individual activities that will enhance understanding.
4.
Provide students with interesting problems and activities that extend the concept of the lesson
5.
Help students develop specific problem solving skills and strategies by providing scaffolded guiding questions
49
Peer tutoring
1.
Team up stronger math skills with lower math skills
Use of manipulative
1.
Eno or smart boards
2.
Dry erase markers
3.
Reference sheets created by special needs teacher
4.
Pairs of students work together to make word cards for the chapter vocabulary
5.
Use 3D shapes for visual learning
6.
Reference sheets for classroom
7.
Graphing calculators
APPENDIX
(Teacher resource extensions)
1.
CCSS. Mathematical Practices:
MP1: Make
sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry
points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and
50
meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try
special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change
course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window
on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations,
verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.
Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students
check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the
approaches of others to solving complex problems and identify correspondences between different approaches.
MP2: Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary
abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it
symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the
ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.
Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the
meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
MP3: Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing
arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze
situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others,
and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from
which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct
logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments
using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are
51
not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades
can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
MP4: Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the
workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply
proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a
design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply
what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision
later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables,
graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their
mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served
its purpose.
MP5: Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and
paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry
software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of
these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school
students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels
are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems.
They are able to use technological tools to explore and deepen their understanding of concepts.
52
MP6: Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in
their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are
careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately
and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give
carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of
definitions.
MP7: Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and
seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.
Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression
x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and
can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 –
3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
MP8: Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper
elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have
a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x +
1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a
problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the
53
reasonableness of their intermediate results.
2. Kuta G 1: Kuta Software – Geometry (Free Worksheets)
3. Teacher Edition: Geometry Common Core by Pearson
4. Student Companion: Geometry Common Core by Pearson
5. Practice and Problem Solving Workbook: Geometry Common Core by Pearson
6. Teaching with TI Technology: Pearson Mathematics by Pearson
7. Progress Monitoring Assessments: Geometry Common Core by Pearson
8. http://www.jmap.org/JMAP_RESOURCES_BY_TOPIC.htm#Geo
9. http://www.mathopenref.com/
10.
http://www.mathisfun.com/
11.
http://www.mathwarehouse.com/
12.
http://www.regentsprep.org/Regents/math/geometry/math-GEOMETRY.htm
13.
http://www.cpm.org/pdfs/state_supplements
14.
http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqgeomapr15.pdf
15.
http://illuminations.nctm.org
16.
http://www.state.nj.us/education/cccs/standards/9/
Notes to teacher (not to be included in your final draft):
54
4 Cs
Three Parts Objective
Creativity: projects
Behavior
Critical Thinking: Math Journal
Condition
Collaboration: Teams/Groups/Stations
Demonstration of Learning (DOL)
Communication – Power points/Presentations
UNIT 2
Similarity, Proof And Trigonometry
Total Number of Days: 42 days
ESSENTIAL QUESTIONS
17.
Grade/Course: __Geometry 10th grade__
ENDURING UNDERSTANDINGS
How do you identify corresponding parts of congruent 23.
If two triangles are congruent, then every pair of their corresponding parts is also
triangles?
congruent.
55
18.
19.
How do you show that two triangles are congruent?
How do you use proportions to find side lengths in
similar polygons?
20.
How do you show two triangles are similar?
24.
Two ways triangles can be proven to be congruent are by using three pairs of
corresponding sides or by using two pairs of corresponding sides and the pair of corresponding
angles included between those sides.
25.
Two geometric figures are similar when corresponding lengths are proportional and
corresponding angles are congruent.
21.
How do you find a side length or angle measure in a
right triangle?
26.
Ratios and proportions can be used to prove whether two polygons are similar and to
find unknown side lengths
22.
How do trigonometric ratios relate to similar right
triangles?
27.
If the lengths of any two sides of a right triangle are known, the length of the third side
can be found by using the Pythagorean Theorem.
28.
Ratios can be used to find side lengths and angle measures of a right triangle when
certain combinations of the side lengths and angle measures are known
RESOURCES
STANDARDS
PACE
CONTENT
SKILLS
(CCSS/MP)
OTHER
Pearson
LEARNING
ACTIVITIES/ASSESSMENTS
(e.g., tech)
To Relate Parallel and Perpendicular G.MG.3 Apply
Text book
geometric
methods
page 164lines
to solve design
170
Example:
problems (e.g.,
3.4
1 day
Parallel and
Perpendicular
lines
designing an object
or structure to
satisfy physical
constraints or
minimize cost;
working with
typographic grid
systems based on
ratios).
Teacher made power points Page 170 Concept Byte –
presentations.
Perpendicular Lines and Plane
http://www.summit.k12.co. Page 167: 1 - 5
us/cms/lib04/CO01001195/
Centricity/Domain/565/cgp0
3gad.pdf
29.
Basic – Problems 1-2
Ex. 6 – 9, 12 – 16 even, 17 – 18, 3139
56
http://www.mathsisfun.com 30.
Average – Problems 1 – 2
/algebra/line-parallelEx. 6 – 18 , 31 –39
perpendicular.html
MP 1
MP 3
31.
Advanced – Problems 1-2
Ex 6 -26, 31 - 39
To use properties of Parallel Lines
G.MG.3
See below
Text book
page 167
http://www.regentsprep.org Standardized Test Prep
/Regents/math/ALGEBRA/A
Q. 27 - 30
C3/Lparallel.htm
Question:
Basic Skills
Review
30 mins PARCC/HSPA
PREP
Using the given information, state the
theorem that allows you to conclude that
j || k
http://www.mathsisfun.com
/algebra/line-parallelperpendicular.html
57
Examine and identify that two
figures are congruent and identify
their corresponding parts
Example:
In the diagram of
4.1 Congruent
2 days
Figures
below,
,
and
and
.
G.SRT.5 Use
Text book
congruence and page 218 similarity criteria 224
for triangles to
solve problems
and to prove
relationships in
geometric figures.
MP 1
PowerGeometry.com
Interactive Practice games
Page 221: 1 - 7
Teacher made power points Building congruent triangles activity
page 225
Practice problem solving exercises
Mathopenref.com/tocs/con
page 222 – 224.
gruencetoc.html
32.
Basic – Problems 1-2
Ex. 8 – 29, 50 – 61
http://jmap.org/htmlstandar
d/Geometry/Informal_and_ 33.
Average – Problems 1 – 4
Formal_Proofs/G.G.28.htm
Ex. 9 – 29 odd, 50 –61
MP 3
MP 4
MP 6
34.
Advanced – Problems 1-4
Ex 9-29 odd, 50-61
Basic Skills
Review
30 mins PARCC/HSPA
PREP
To find angle measures of a triangle
using Angle Sum Theorem
G.SRT.5
See below
Text book
page 224
Mathopenref.com/tocs/con
gruencetoc.html
Page 224: Standardized Test
Prep – Q. 50 - 53
Question:
The measure of one angle In a triangle is
80˚
r w ang ar c ngru n
What is the measure of each?
58
Conclude and defend that two
triangles congruent using the SSS
and SAS Postulates
SSS stands for "side, side, side" and
means that we have two triangles
with all three sides equal.
G.SRT.5 Use
Text book
congruence and page 226 similarity criteria 230
for triangles to
solve problems
and to prove
relationships in
geometric figures.
SAS stands for "side, angle, side" and
4.2 Triangle
MP 1
2 days Congruence by means that we have two triangles
where we know two sides and the
SSS and SAS
MP 3
included angle are equal.
PowerGeometry.com
35.
Teacher made power points
36.
37.
38.
http://mathopenref.com/co
ngruentsss.html
39.
http://www.mathsisfun.com
/geometry/congruent.html
40.
Page 230:1-7
Lesson Quiz
Practice and Problem
Solving Exercises page 230
Practice and Problem
Solving Exercises
Basic – Problems 1-2
Ex. 8 – 12, 35– 46
Average – Problems 1 – 2
Ex. 9, 35 –46
41.
Advanced – Problems 1-3
Ex 9-13 odd, 15- 46
Basic Skills
Review
30 mins PARCC/HSPA
PREP
To Find the coordinates of one
endpoint using the midpoint
formula.
G.C0.1
Question:
Find the point on a
directed line
segment between
A segment has a midpoint at (2, 2)
G.GPE.6
Text book
page 233
http://jmap.org/htmlstand 42.
Standardized Test Prep
ard/Geometry/Coordinate
Questions 35 – 38
_Geometry/G.G.67.htm
59
and an endpoint at (-2, 4). What are two given points
the coordinates of the other endpoint that partitions the
segment in a given
of the segment?
ratio.
Construct a proof defending that
two triangles congruent using the
ASA and AAS Postulates
G.SRT.5 Use
Text book
congruence and page 234 similarity criteria 240
for triangles to
ASA an f r "ang , i , ang ”
solve problems
and means that we have two
and to prove
triangles where we know two angles
relationships in
and the included side are equal.
geometric figures.
4.3 Triangle
2 days Congruence by
ASA and AAS
For example: the 2 triangles below
are congruent
MP 1
* PowerGeometry. Com
44.
Page 238: 1 - 7
* Teacher made power
point presentations
45.
Lesson Quiz page 241A
46.
Basic – Problems 1-2: Ex. 8
43.
http://www.mathsisf
– 12, 32- 39
un.com/geometry/congru
47.
Average – Problems 1 – 2:
ent.html
Ex. 9 – 11 odd, 32-39
48.
Advanced – Problems 1-2:
Ex 9-11 odd, 32 – 39
MP 3
AAS stands for "angle, angle, side"
and means that we have two
triangles where we know two angles
and the non-included side are equal.
60
For example: the 2 triangles below
are congruent
To write converses, inverses and
contrapositives of conditionals.
Basic Skills
Review
30 mins PARCC/HSPA
PREP
1 day
Question:
ri
c nv r
f “ If y u ar
than 18 years old, then you are too
y ung v
in
ni
Sa ”
To prove right triangles congruent
4.6 Congruence using the Hypotenuse Leg Theorem
in Right
Triangles
Example:
Prepares for
G.CO.11 Prove
Text book
page 241
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.SRT.5 Use
Text book
congruence and page 258 similarity criteria 264
for triangles to
solve problems
http://www.cpm.org/pdfs/st Standardized Test Prep Questions
ate_supplements/Logical_St 32 - 35
atements.pdf
http://www.finneytown.or
g/Downloads/GETE04062.p
Page 261: 1 - 7
df
61
and to prove
relationships in
geometric figures.
49.
http://www.mathopenref.co Ex. 8 – 16, 20,22,25,32 - 36
m/congruenthl.html
50.
Average – Problems 1 – 2
MP 1
Ex. 9 , 11– 28, 32 –36
MP 3
51.
Question:
Basic Skills
Review
30 mins PARCC/HSPA
PREP
Advanced – Problems 1-2
http://www.regentsprep.org
/Regents/math/geometry/G Ex 9, 11-28, 32 - 36
P4/Ltriangles.htm
Solution
To determine whether given
triangles are congruent.
Basic – Problems 1-2
G.SRT.5
See above
Text book
page 264
http://www.regentsprep.o 52.
Standardized Test Prep
rg/Regents/math/geometr
Questions 29 – 31
y/GP4/Ltriangles.htm
Determine whether the triangles are
congruent. If they are, write a
congruent statement.
62
To identify congruent overlapping
triangles, prove two triangles
congruent using other congruent
triangles
For example:
G.SRT.5 Use
Text book
congruence and page 265 –
similarity criteria 271
for triangles to
solve problems
and to prove
relationships in
geometric figures.
Separate and redraw DFG and
4-7
2days
EHG. Identify the common angle.
MP 1
Congruence in
Overlapping
Triangles
30 mins PARCC/HSPA
PREP
Page 268: 1 - 7
Basic – Problems 1-2:
Ex. 8 – 16, 17 – 20, 33 - 37
http://www.youtube.com/w
55.
atch?v=wlEYuPhShig
Average – Problems 1 – 2:
Ex. 9– 15 odd, 17 , 19 -26, 33 37
56.
Teacher made power point
presentation
Advanced – Problems 1-2:
Ex 9-15 odd, 17, 19 – 28, 33 - 37
MP 3
http://www.sophia.org/over
lappingtriangles/overlappingtriangles--2tutorial?topic=congruenttriangles
Solution
Basic Skills
Review
http://www.finneytown.org
53.
/Downloads/GETE04072.pdf
54.
To prove congruent segments in
overlapping triangles.
Question:
G.SRT.5
See above
Text book
page 271
http://www.sophia.org/over 57.
Standardized Test Prep
lappingQuestions 29 – 32
triangles/overlappingtriangles--2tutorial?topic=congruenttriangles
63
To use properties of midsegments to G.CO.10
solve problems
G.CO.12
For example:
1
day
5-1
Midsegments of
Triangles
G.SRT.5 Use
congruence and
similarity criteria
for triangles to
solve problems
and to prove
relationships in
geometric figures.
Text book http://www.finneytown.org
58.
page 284 – /Downloads/GETE05012.pdf
291
59.
60.
http://www.regentsprep.org
/Regents/math/geometry/G
61.
P10/MidLineL.htm
62.
http://www.mathopenref.co
m/trianglemidsegment.html
Page 284 Concept Byte –
Investigating Midsegments
Page 288: 1 - 6
Basic – Problems 1-3:
Ex. 7 – 26, 28 – 30, 32 – 42 even
Average – Problems 1 – 3:
Ex. 7– 25 odd, 26 -45,
Advanced – Problems 1-3:
Ex 7-25 odd, 26 – 48, 53 - 57
MP 1
MP 3
MP 5
64
To find the midsegment of a triangle G.CO.10
Question:
G.CO.12
Text book
page 291
http://www.regentsprep.org 63.
Standardized Test Prep
/Regents/math/geometry/G
Questions 49 - 52
P10/MidLineL.htm
G.SRT.5
See above
Basic Skills
Review
30 mins PARCC/HSPA
PREP
1. The triangular face of the rock and
Roll Hall of Fame in Cleveland, Ohio is
isosceles. The length of the base is
229ft 6in. What is the length of the
highlighted segment?
2. Explain your reasoning.
65
To use properties of perpendicular
bisectors and angle bisectors
G.CO.9
5-2
G.SRT.5 Use
congruence and
similarity criteria
for triangles to
solve problems
and to prove
relationships in
geometric figures.
Perpendicular
and Angle
Bisectors
MP 1
For example:
Text book http://163.150.89.242/YHS/
64.
page 292 – Faculty/AB/bagg/Geometry/
images/Geometry%20text%
299
65.
20PDFs/5.6.pdf
66.
http://www.youtube.com/w
67.
atch?v=wxsr8egcq0M
68.
http://www.mathopenref.co
m/bisectorperpendicular.ht
ml
1
day
Page 284 Concept Byte –
Investigating Midsegments
Page 288: 1 - 6
Basic – Problems 1-3:
Ex. 7 – 26, 28 – 30, 32 – 42 even
Average – Problems 1 – 3:
Ex. 7– 25 odd, 26 -45,
Advanced – Problems 1-3:
Ex 7-25 odd, 26 – 48, 53 - 57
MP 3
MP 5
Basic Skills
Review
30 mins PARCC/HSPA
PREP
To construct perpendicular bisector G.CO.10
of a triangle
G.CO.12
Question:
G.SRT.5
Text book
page 299
http://www.mathopenref.co 69.
Standardized Test Prep
m/bisectorperpendicular.ht
Questions 39 - 42
ml
See above
66
A company plans to build a
warehouse that is equidistant from
each of its three stores, A, B, and C.
Where should the warehouse be
built?
Discriminate - interior and exterior
angles of polygons and their sums.
Determine and justify angle
measures using Polygon Angle-Sum
Theorems
6-1
2days
The Polygon
Angle-Sum For example:
Theorems
Find the number of degrees
in each interior angle of a regular
dodecagon.
It is a regular polygon, so we can use
the formula.
In a dodecagon, n = 12.
G.SRT.5 Use
Text book
congruence and page 353 similarity criteria 358
for triangles to
solve problems
and to prove
relationships in
geometric figures.
MP 1
http://mathopenref.com/co
70.
ngruentsss.html
71.
http://www.regentsprep.org
72.
/Regents/math/geometry/G
G3/LPoly2.htm
73.
Page 356: 1 - 6
Basic – Problems 1-2:
Ex. 7 – 14, 49 - 54
Average – Problems 1 – 2:
Ex. 7 – 13 odd, 22 – 25, 49 -54
Advanced – Problems 1-4:
Ex 7-21 odd, 22 – 44, 49 -54
Activity on Exterior Angles of
polygon page 352
MP 3
67
To find the sum of the measures of
interior and exterior angles of
polygon.
Question:
Basic Skills
Review
30 mins PARCC/HSPA
What is m∠ x ?
G.SRT.5 Use
Test boot
congruence and page 358
similarity criteria
for triangles to
solve problems
and to prove
relationships in
geometric figures.
http://www.cde.ca.gov/ta/t Standardized Test Prep Questions
g/sr/documents/cstrtqgeom 45 - 48
apr15.pdf
PREP
2 days
Determine and justify sides and
Text book
G.SRT.5
angles through relationships among
page 359 G.CO.11
Prove
6-2
parallelograms
366
theorems about
Properties of For example:
parallelograms.
Parallelograms
Theorems include:
opposite sides are
In the accompanying diagram of
congruent,
http://jmap.org/htmlstandar
74.
d/Geometry/Informal_and_
75.
Formal_Proofs/G.G.38.htm
Page 363: 1 - 8
Basic – Problems 1-4:
Ex. 9 – 24, 38- 41, 49- 54
76.
Average – Problems 1 – 4:
Ex. 9 – 23 odd, 25 - 41, 49 -54
77.
Advanced – Problems 1-4:
68
parallelogram ABCD,
and
degrees in
. Find the number of
.
opposite angles are
congruent, the
diagonals of a
parallelogram
bisect each other,
and conversely,
rectangles are
parallelograms with
congruent
diagonals.
Ex 9-23 odd, 25 – 44, 49 -54
MP 1
MP 3
Basic Skills
Review
30 mins PARCC/HSPA
PREP
To apply the relationship among
sides, angels and diagonals of
parallelograms.
G.CO.11 See aboveText book
page 366
http://www.cde.ca.gov/ta/t Standardized Test Prep Questions
g/sr/documents/cstrtqgeom 45 - 48
apr15.pdf
Question:
What values of a and b make
quadrilateral MNOP a parallelogram?
69
1 day
G.SRT.5
Justify that a quadrilateral is a
Test book
parallelogram using the properties of
page 367 G.CO.11 Prove
parallelogram
374
theorems about
parallelograms.
Theorems include:
For example:
opposite sides are
congruent, opposite
The accompanying diagram shows
6-3
angles are
quadrilateral BRON, with diagonals congruent, the
Proving that a
and
, which bisect each other diagonals of a
Quadrilateral Is
parallelogram bisect
at X.
a Parallelogram
each other, and
conversely,
rectangles are
parallelograms with
congruent diagonals.
http://jmap.org/htmlstandar
78.
d/Geometry/Informal_and_
79.
Formal_Proofs/G.G.27.htm
Page 372: 1 - 6
Basic – Problems 1-3:
Ex. 7 – 16, 18- 20, 22- 28, 32-44
80.
Average – Problems 1 – 3:
Ex. 7 – 15, odd, 17 - 28, 32 - 44
81.
Advanced – Problems 1-3:
Ex 7-15 odd, 17 – 28, 32 -44
Prove: MP 1, MP 3
To determine whether a
quadrilateral is a parallelogram
Basic Skills
Review
30 mins PARCC/HSPA
PREP
Question:
Based on the markings, determine if
the figure is a parallelogram. If so,
justify your answer.
Text book
page 374
G.CO.11 See above
G.SRT.5
http://www.cde.ca.gov/ta/t Standardized Test Prep Questions
g/sr/documents/cstrtqgeom 29 - 31
apr15.pdf
http://www.jmap.org/Static
Files/PDFFILES/WorksheetsB
yPI/Geometry/Informal_and
_Formal_Proofs/Drills/PR_G.
G.38_2.pdf
70
I day
Analyze parallelograms to determine G.SRT.5
Text book
page 375 special types
G.CO.11 Prove
382
theorems about
parallelograms.
For example:
Theorems include:
opposite sides are
Rectangle:
congruent,
parallelogram
opposite angles are
6-4
with 4 right angles
congruent, the
Properties of Rhombus:
diagonals of a
Rhombus, parallelogram
parallelogram
Rectangle, with all 4 sides
bisect each other,
Square
congruent
and conversely,
rectangles are
Square:
parallelograms with
a rectangle with
congruent
all 4 sides congruent
diagonals.
http://www.regentsprep.org
82.
/Regents/math/geometry/G
83.
P9/LRectangle.htm
Page 379: 1 - 6
Basic – Problems 1-3:
Ex. 7 – 23, 24 – 40 even, 41, 43,
46, 47, 60 -69.
84.
Average – Problems 1 – 3:
Ex. 7 – 23, odd, 24 - 54, 60 - 69
85.
Advanced – Problems 1-3:
Ex 7-23 odd, 24 – 54, 60-69
MP 1
MP 3
71
To find a side length of a
parallelogram
Basic Skills
Review
Text book
page 382
G.CO.11 See above
G.SRT.5
Question:
What is the height of this rectangle?
http://www.jmap.org/Static
Files/PDFFILES/WorksheetsB
yPI/Geometry/Informal_and
_Formal_Proofs/Drills/PR_G.
G.38_2.pdf
30 mins PARCC/HSPA
PREP
Determine if a parallelogram is a
Rhombus, Rectangle or Square
G.SRT.5
Text book
page 383 388
G.CO.11 Prove
theorems about
Conditions for
parallelograms.
2 days
Rhombus, For example:
Theorems include:
Rectangle,
Which reason could be used to prove opposite sides are
Square
that a parallelogram is a rhombus? congruent,
opposite angles are
congruent, the
6-5
http://www.cde.ca.gov/ta/t Standardized Test Prep Questions
g/sr/documents/cstrtqgeom 55 - 58
apr15.pdf
http://www.jmap.org/htmls
86.
tandard/Geometry/Informal
87.
_and_Formal_Proofs/G.G.39
.htm
Page 386: 1 - 7
Basic – Problems 1-3:
Ex. 8 – 18, 24 – 31, 36 -43
88.
Average – Problems 1 – 3:
Ex. 9 – 13, odd, 15 - 31, 36 - 43
89.
Advanced – Problems 1-3:
Ex 9 -13 odd, 15 – 31, 36 - 43
72
1) Diagonals are congruent.
2) Opposite sides are parallel.
3) Diagonals are perpendicular.
4) Opposite angles are
congruent.
diagonals of a
parallelogram
bisect each other,
and conversely,
rectangles are
parallelograms with
congruent
diagonals.
MP 1
MP 3
Basic Skills
Review
30 mins PARCC/HSPA
PREP
To determine whether a given
parallelogram is a rhombus or
rectangle
Text book
page 388
G.CO.11 See above
G.SRT.5
http://www.cde.ca.gov/ta/t Standardized Test Prep Questions
g/sr/documents/cstrtqgeom 32- 35
apr15.pdf
Question:
Each diagonal of a quadrilateral
bisects a pair of opposite angles of
the quadrilateral. What is the most
http://www.jmap.org/Static
Files/PDFFILES/WorksheetsB
yPI/Geometry/Informal_and
_Formal_Proofs/Drills/PR_G.
73
precise name for the quadrilateral?
1.
parallelogram
2.
rhombus
3.
rectangle
4.
not enough information
To determine, verify and apply the G.SRT.5 Use
Text book
properties of trapezoids and Kite to congruence and page 389 similarity criteria 397
solve problems.
for triangles to
Example:
solve problems
and to prove
Exa : In the diagram below of isosceles
relationships in
trapezoid DEFG,
,
geometric figures.
,
,
, and
. Find the value of
6-6
x.
MP 1
2 days Trapezoids and
Kites
MP 3
G.38_2.pdf
http://www.jmap.org/htmls
5.
tandard/Geometry/Informal
6.
_and_Formal_Proofs/G.G.40
.htm
Page 393: 1 - 6
Basic – Problems 1-4:
Ex. 7 – 24, 26 – 34 even, 46 –
49, 71 - 76
7.
Average – Problems 1 – 4:
Ex. 7 – 23, odd, 25 - 62, 71- 76
8.
Advanced – Problems 1-4:
Ex 7 -23 odd, 25 – 66, 71 - 76
74
To verify and apply properties of
trapezoids and kites
G.SRT.5
See above
Question:
Text book
page 397
http://www.cde.ca.gov/ta/t Standardized Test Prep Questions
g/sr/documents/cstrtqgeom 67 - 70
apr15.pdf
Figure ABCD is a kite.
http://www.jmap.org/Static
Files/PDFFILES/WorksheetsB
yPI/Geometry/Informal_and
_Formal_Proofs/Drills/PR_G.
G.38_2.pdf
Basic Skills
Review
30 mins PARCC/HSPA
PREP
What is the area of figure ABCD, in
square centimeters?
7.1
2 days Ratios and
Proportions
Conclude from evidence provided
which sides correspond in similar
triangles and identify appropriate
ratios to establish proportions and
solve for a missing side length
For example:
If ∆ABC ∼∆ E ,
n i n ify
appropriate ratios, establish a
G.SRT.5 Use
Text book 9.
PowerGeometry.Co
congruence and page 432 m
similarity criteria 438
10.
Teacher made
for triangles to
power points
solve problems
presentations
and to prove
relationships in
geometric figures.
MP 1
11.
Page 436: 1 - 8
12.
Test prep page 438
13.
Basic – Problems 1-3:
Ex. 9 – 16, 61- 69
14.
Average – Problems 1 – 3:
Ex. 9 – 15 odd, 33-34, 61-69
15.
Advanced – Problems 1-3:
Ex 9-15 odd, 33- 34, 61-69
75
proportion and solve for side length MP 3
x.
MP 7
Basic Skills
Review
30 mins PARCC/HSPA
PREP
To apply ratio and proportion to
solve real life problem
Prepares for
Test book
G.SRT.5 See below page 438
Standardized Test Prep: Questions
61 - 65
Questions:
The ratio of the width to the height of
i ’ c
u r
ni r cr n i 6:
If the screen is 12 inches high, how
wide is it?
76
2 days
7.2 Similar
Polygons
Examine similar polygons and utilize G.SRT.5 Use
Text book
traits of similar polygons to solve
congruence and page 440 similarity criteria 446
problems
for triangles to
solve problems
and to prove
For example:
relationships in
geometric figures.
http://jmap.org/htmlstandar
d/Geometry/Informal_and_
Formal_Proofs/G.G.45.htm 16.
If
19.
Advanced – Problems 1-2:
Teacher-made Power Point
Ex 9-17 odd, 51-64
presentation
,
, and
length of
,
. What is the
?
,
MP 1
Page 444: 1 - 8
17.
Basic – Problems 1-2:
Ex. 9 – 17, 51- 64
http://mathopenref.com/si
18.
Average – Problems 1 – 2:
milarpolygons.html
Ex. 9 – 17 odd, 51-64
MP 3
Basic Skills
Review
30 mins PARCC/HSPA
PREP
To apply scale factor to find the
length of a segment
G.SRT.5 Use
Text book
congruence and page 447
similarity criteria
Question:
for triangles to
∆P S ~ ∆JKL wi a ca fac r f solve problems
4:3, QR = 8cm. What is the value of KL?and to prove
relationships in
geometric figures.
http://jmap.org/htmlstandar Standardized Test Prep: Questions
d/Geometry/Informal_and_ 51 - 54
Formal_Proofs/G.G.45.htm
77
To apply the AA Similarity Postulate and G.SRT.5 Use
the SAS and SSS Similarity Theorems.
congruence and
To use the similarity to determine and
justify indirect measurements.
For example:
2 days
Text book
page 450 similarity criteria 458
for triangles to
solve problems
and to prove
relationships in
geometric figures.
Given that the triangles below are similar
- If two of their angles are equal, then the
third angle must also be equal, because
angles of a triangle always add to 180. In G.GPE.5
7.3 Proving
Triangle Similar this case the missing angle is 180 - (72 + MP 1
35) = 73
MP 3
20.
Page 455: 1 - 6
http://jmap.org/htmlstandar
21.
d/Geometry/Informal_and_
Formal_Proofs/G.G.45.htm
22.
Basic – Problems 1-2:
Ex. 7 – 12, 37 - 52
http://mathopenref.com/si
23.
milarpolygons.html
Advanced – Problems 1-4:
Ex 7-17 odd, 18 - 52
Average – Problems 1 – 2:
Ex. 7 – 11 odd, 37 - 52
Teacher-made Power Point
presentations
MP 3
78
Basic Skills
Review
30 mins PARCC/HSPA
PREP
To apply the Pythagorem Theorem
to find the missing side of a right
triangle
Question:
G.SRT.8 Use
Text book
trigonometric
page 458
ratios and the
Pythagorean
Theorem to solve
right triangles in
applied problems.
http://jmap.org/htmlstandar
24.
d/Geometry/Informal_and_
Formal_Proofs/G.G.48.htm
25.
Standardized Test Prep:
Questions 37- 40
Worksheet from the stated
website
A 17 ft ladder leans against a wall, if
the ladder is 8ft from the base of the
wall, how far is it from the bottom of
the wall to the top of the ladder.
79
UNIT #3
Extending to three dimensions
80
Grade/Course: __Geometry 10th grade__
Total Number of Days: 19 days
ESSENTIAL QUESTIONS
26.
27.
ENDURING UNDERSTANDINGS
How do you find the area of a polygon or 28.
find the circumference and area of a
circle?
29.
How can you determine the intersection
of a solid and a plane?
30.
The area of a regular polygon is a function of the distance from the center to a side and the
perimeter.
A three-dimensional figure can be analyzed by describing the relationships among its vertices, edges,
and faces.
The surface area of a three-dimensional figure is equal to the sum of the areas of each surface of the
figure.
RESOURCES
STANDARDS
PACING
CONTENT
SKILLS
LEARNING ACTIVITIES and ASSESSMENTS
OTHER
(CCSS/MP)
Pearson
(e.g., tech)
10.1
Areas of
parallelograms
and triangles
2 days
To find the area of
parallelograms and
triangles.
G.MG.1 Use
geometric shapes,
their measures, and
their properties to
Example:
describe objects (e.g.,
modeling a tree trunk
The piece of stained glass
or a human torso as a
at the bottom is made up
cylinder)
of eight congruent
parallelograms. Each
parallelogram has a base
of
8 centimeters and a height
of 3 centimeters. Find the
MP.3
Pearson
Chapter
#10.1 Get
Ready
Web link for the area
of parallelograms and
1.
triangles:
1) problems 1-2
www.wyzant.com/help/math/.../a
reas/parallelograms_and_tria
Exs 8-13 all, 47-62
ngles
Text book
page #616622
2) Problems 3-4
www.virtualnerd.com/geometry/l
ength-area/parallelogramtriangles-area
Kuta software for
worksheet:
Basic
Exs. 14-17 all, 19, 22, 23,
30-40 even, 37-38
2.
Average:
www.kutasoftware.com/.../6-
81
area of the entire piece.
MP.5
Area%20of%20Triangles%20a
1) problems 1-2
Exs. 9-13 odd, 47-62.
www.kutasoftware.com/.../Area
%20of%20Squares,%20Rectan
2) Problems 3-4
Exs. 15-17 odd, 18-43
you tube video tutorial:
www.youtube.com/watch?v=Fac
www.youtube.com/watch?v=FG
LWKWcg0Vo
3.
Advanced:
1) Problems 1-4
Exs. 9-17 odd, 18-62
Math power point notes:
Google search:
granicher.wikispaces.com/.../b)+Area+of+a+Parallelog
ram+%26+Triangl..
Math skills practice:
www.finneytown.org/Downloads/wk10.pdf
82
Basic
skills Apply distance in
Use
Review:
the coordinate
plane.
Ex
Example:
HSPA
PREP/PAR
CC/SAT
G.GPE.6
Pearson
Find the point on a
Chapter #10
directed line segment Get Ready
between two given
points that partitions
the segment in a given
ratio.
Kuta software for
worksheets
Text book
page # 622
Standardized Test Prep
book page 622 Q. 47-49
(SAT/HSPA) Text
www.kutasoftware.co
m/.../Area%20of%20S
quares,%20Rectangles
30 mn
Ex
On the directed line
segment from R to S on
the coordinate plane
above, what are the
coordinates of the point
that partitions the
segment in the ratio 2 to
3?
83
84
10.2
Areas of
trapezoids,
rhombuses,
and kites
Pearson
Find the area of a G.MG.1
trapezoid,
Use geometric
Chapter
rhombus, and kite.
shapes, their
#10.2 Get
measures, and their Ready
Example:
properties to describe
The roof on the bridge
objects (e.g., modeling
below consists of four
a tree trunk or a
sides, two congruent
Text book
human torso as a
trapezoids and two
page #623cylinder)
congruent triangles.
628
MP.1- 6
Web link for the area 1.
of a trapezoid,
rhombus, and kite:
2) Problems 3-4
Exs. 20-25 all, 26-38 even
www.khanacademy.org/.../area..
./areas_of_trapezoids_rhombi
_and_kites
2.
-Use the diagram above to
find the combined area of
the two triangles.
-What is the area of the
entire roof?
Average:
1) problems 1-2
Kuta software for
worksheet:
www.mybookezz.org/kutasoftware-infinite-geometryfinding-total-area
-Find the combined area of
the two trapezoids.
1) problems 1-2
Exs 11-19 all, 45-53
www.slideshare.net/.../112areas-of-trapezoidsrhombuses-and-kites
Exs. 11-19 odd, 45-53.
2) Problems 3-4
www.kutasoftware.com/FreeW
orksheets/.../Area%20of%20Tra
pezoids
2 days
Basic:
Exs. 21-25 odd, 26-41
3.
Advanced:
1) Problems 1-4
you tube video tutorial:
Exs. 11-25 odd, 26-53
www.youtube.com/watch?v=1N
DXo8nnRUE
Math power point notes:
www.youtube.com/watch?v=V2x Google
-
search:
nehsmath.wikispaces.com/.../74+PPT+Areas+of+Trapezoids,+Rhombus
Math skills practice:
85
www.khanacademy.org/.../area.../areas_of_trapezoids
_rhombi_and_kites
86
Basic skills
Review:
Apply distance in
HSPA
the coordinate
PREP/PARCC/S plane.
AT
Example:
G.GPE.6
Pearson
Find the point on a
Chapter #10
directed line segment Get Ready
between two given
points that partitions
the segment in a given
ratio.
Kuta software for
worksheets
Text book
page # 628
30 mn
www.kutasoftwar
e.com/FreeWorks
heets/GeoWorks
heets/3Points%20in...
Standardized Test Prep
(SAT/HSPA)
Text book page 628 Q. 45-47
87
10.3
Areas of
Regular
Polygons
Find the area of a
regular polygon.
G.MG.1
Use geometric
Chapter
shapes, their
#10.3 Get
Example:
measures, and their Ready
The gazebo in the photo is properties to describe
built in the shape of a
objects (e.g., modeling
regular octagon. Each side a tree trunk or a
Text book
is 8 ft long, and its
human torso as a
page #629apothem is 9.7 ft. What is cylinder)
634
the area enclosed by the
gazebo?
MP.1
1 day
Pearson
MP.3
Web link for the area 4.
of a regular polygon:
www.mathwords.com/a/area_re
gular_polygon.htm
Basic:
1) problems 1-3
Exs 8-25 all, 26-30 even
31-33 all, 35, 44-52
www.kutasoftware.com/.../6Area%20of%20Regular%20Pol 5.
ygons.pdf
1) problems 1-3
Kuta software for
worksheet:
Exs. 9-25 odd, 26-41, 44-52
6.
www.kutasoftware.com/FreeW
orksheets/.../Area%20of%20Tra
pezoids
Average:
Advanced:
1) Problems 1-3
Exs. 9-25 odd, 26-52
MP.4
MP.7
www.gobookee.net/kutasoftware-area-of-regularpolygons-answ
Math
power point notes:
Google search:
you tube video tutorial: jcs.k12.oh.us/teachers/.../PH_Geo_103_Areas_of_Regular_Polygons.pp
www.youtube.com/watch?v=eQ
hgozrRiYI
Math skills practice:
www.youtube.com/watch?v=Hlc www.finneytown.org/Downloads/GETE1003.pdf
Hd-psOWs
88
Basic skills
Review:
30 mn
Partition line
segment given
ratio.
HSPA
PREP/PARCC/S Example:
AT
Point p lies on the
direct line segment
from A(2,3) to
B(8,0)and partitions
the segment in the
ratio 2 to 1. What
are the coordinates
of point P?
G.GPE.6
Pearson
Kuta software for
worksheets
Find the point on a
Chapter #10
directed line segment Get Ready schoolwires.henry.k12.
between two given
ga.us/.../415_Partitioning%20a
points that partitions
the segment in a given
ratio.
Standardized Test Prep
(SAT/HSPA)
Text book page 634 Q. 44-47
Text book
page # 634
89
10.4
Perimeters
and areas of
similar figures
Find the
perimeters and
areas of similar
polygons.
Community Service During the
summer, a group of high school
students used a plot of city land
and harvested 13 bushels of
vegetables that they gave to a food
pantry. Their project was so
successful that next summer the
city will let them use a larger,
similar plot of land.
In the new plot, each dimension is
2.5 times the corresponding
dimension of the original plot.
G.MG.3
Pearson
Apply geometric
Chapter
methods to solve
#10.3 Get
design problems (e.g., Ready
designing an object or
structure to satisfy
physical constraints or
minimize cost; working Text book
with typographic grid page #629634
systems based on
Web link for finding
the perimeters and
areas of similar
polygons:
7.
1) problems 1-2
Exs 9-16 all, 52-62
www.onemathematicalcat.org/M
ath/...obj/per_area_similar_figu
2) Problems 3-4
res.htm
Exs. 17-24 all, 26-30 even
31-33 all, 34-44 even
www.kutasoftware.com/freeige.
8.
Average:
ratios).
Kuta software for
worksheet:
MP.1
1) problems 1-2
Exs. 9-15 odd, 52-62.
www.kutasoftware.com/freeige.
2) Problems 3-4
2 days
MP.5
MP.8
you tube video tutorial:
Exs. 17-23 odd, 25-47
www.youtube.com/watch?v=ae-
9.
Advanced:
1) Problems 1-4
www.youtube.com/watch?v=ML
How many bushels can they
expect to harvest next year?
Basic:
AdoSrJfi0
Exs. 9-23 odd, 25-62
Math
power point notes:
Google search:
www.villagechristian.org/media/2322216/lesson%2086.ppt
Math skills practice:
90
www.mathwarehouse.com/.../similar/triangles/areaand-perimeter-of-simi
91
Basic skills
Review:
Partition line
segment given
ratio.
G.GPE.6
Pearson
Kuta software for
worksheets
Find the point on a
Chapter #10
directed line segment Get Ready schoolwires.henry.k12.
HSPA
between two given
ga.us/.../4PREP/PARCC/S Example:
15_Partitioning%20a
points
that
partitions
AT
Point R lies on the
the segment in a given
Text book
directed line segment ratio.
page # 641
from L(-8, -10) to
Standardized Test Prep
(SAT/HSPA)
Text book page 641 Q. 52-55
M(4, -2) and
30 mn
partitions the
segment in the ratio
3 to 5. What are the
coordinates of point R?
92
Recognize
polyhedral and
Spaces Figures
their parts.
and Cross
Section
-To visualize cross
sections of space
figures.
11.1
Example:
Julie incorrectly identified
the solid below as a pyramid
with a square base.
1. Correctly identify the
solid.
2 days
2. What would you say to
Julie to help her tell the
difference between this
solid and a
pyramid?
G.GMD.4
Pearson
Identify the shapes of
two-dimensional
cross-sections of
three- dimensional
objects, and identify
three-dimensional
objects generated by
rotations of twodimensional objects.
Chapter
#11.1 Get
Ready
MP.1
MP.3
MP.4
MP.5
Web link for solid
figures:
10.
Basic:
1) problems 1-3
www
› Geometry
Concepts and Skills › Chapter 9
Exs 6-17 all, 51-62
2) Problems 4-5
Text book
page #688695
www.superteacherworksheets.
com/solid-shapes
Kuta software for
worksheet:
Exs. 18-23 all, 24-34 even, 38
11.
Average:
1) problems 1-3
www.kutasoftware.com/FreeW
orksheets/GeoWorksheets/10-
Exs. 7-17 odd, 51-62.
2) Problems 4-5
you tube video tutorial:
Exs. 19-23 odd, 24-40
www.youtube.com/watch?v=DB
S-8meBgZs
12.
MP.7
www.youtube.com/watch?v=q2
Advanced:
1) Problems 1-3
Exs. 7-17 odd, 51-62
Math
power point notes:
Google search:
www.clintweb.net/ctw/ppsspowerpointsolidshapes.pp
t
Math skills practice:
www.ixl.com/math/grade-5/identify-planar-and-solid-
93
figures
94
Basic skills
Review:
Use the Pythagorean
theorem to solve
problems.
HSPA
PREP/PARCC/S Example:
AT
G.SRT.8
Pearson
Use trigonometric ratios Chapter #11
and the Pythagorean
Get Ready
Theorem to solve right
triangles in applied
problems.
Text book
page # 695
30 mn
Kuta software for
worksheets
Standardized Test Prep (SAT/HSPA)
Text book page 695 Q. 51-55
www.gobookee.net/kut
a-software-righttriangles-andpythagorean-theor
Triangle JKL above
represents the boundary
of a state wilderness
area. An access road
will be constructed that
intersects side JK at a
90 degrees angle and
extends to point L. The
road will intersect side
JK 24 miles from point
K, and the length of side
KL is 30 miles.
Part A:
What is the length, in
miles, of the access
road? Show your work.
95
Part B:
What is the length, in
miles, of side JK? Show
your work.
96
97
Find the surface
area of prism and
Surface Areas
cylinder.
of Prisms and
Cylinders
Example:
11.2
G.MG.1
Pearson
Chapter
#11.2 Get
Ready
Use geometric
shapes, their
Architecture In this exercise measures, and their
below use the following
properties to describe
information. Suppose a
objects (e.g., modeling Text book
skyscraper is a prism that a tree trunk or a
page #699is 415 meters tall and each human torso as a
707
base is a square that
cylinder).
measures 64 meters on a
side.
MP.1
2 days
MP.3
MP.7
Web link for finding
the surface area of
prism and cylinder:
13.
Basic:
1) problems 1-2
Exs 7-13 all, 44-45
hotmath.com/help/gt/genericpre
alg/section_9_4.html
2) Problems 3-4
Exs. 14-20 all, 22-30 even, 37
www.virtualnerd.com/geometry/
surface-area.../prismscylinders-area
14.
Kuta software for
worksheet:
Average:
1) problems 1-2
Exs. 7-13 odd, 44-45.
www.kutasoftware.com/.../10Surface%20Area%20of%20Pri
sms
2) Problems 3-4
Exs. 15-19 odd, 21-38
you tube video tutorial:
www.youtube.com/watch?v=DB
1. What is the lateral area
of this skyscraper? 2.
Challenge What is the
surface area of this
skyscraper? (Hint: The
ground is not part of the
surface area of the
skyscraper.)
MP.8
S-8meBgZs
15.
Advanced:
1) Problems 1-4
www.youtube.com/watch?v=q2
Exs. 7-19 odd, 21-55
Math
power point notes:
Google search:
https://www.madison.k12.al.us/.../Surface%20Area%2
0of%20Prisms
Math skills practice:
www.ixl.com/math/grade-8/surface-area-of-prisms-
98
and-cylinders
99
Basic skills
Review:
Use proportions.
G.SRT.5
Example:
Use congruence and
Chapter #11
similarity criteria for
Get Ready
triangles to solve
problems and to prove
relationships in
geometric figures.
Text book
page # 707
HSPA
PREP/PARCC/S
AT
30 mn
The figure above
represents a swing set.
The supports on each
side of the swing set are
constructed from two
12-foot poles connected
by a brace at their
midpoint. The distance
between the bases of
the two poles is 5 feet.
Pearson
Kuta software for
worksheets
Standardized Test Prep (SAT/HSPA)
Text book page 707 Q. 44-47
www.gobookee.net/kut
a-software-righttriangles-andpythagorean-theor
Part A:
What is the length of
each brace?
Part B:
Which theorem about
triangles did you apply
to find the solution in
Part A?
100
101
Find the surface
area of a pyramid
Surface areas
and cone.
and pyramids
and cones
Example:
11.3
G.MG.1
Pearson
Chapter
#11.3 Get
Ready
Use geometric
shapes, their
Veterinary Medicine A cone- measures, and their
properties to describe
shaped collar, called an
Elizabethan collar, is used to objects (e.g., modeling Text book
a tree trunk or a
page #708prevent pets from
aggravating a healing wound. human torso as a
715
cylinder).
2 days
Web link for finding 3.
the surface area of a
pyramid and cone:
Basic:
1) problems 1-4
Exs 9-15 all, 44-53
www.virtualnerd.com/geo
metry/surfacearea.../pyramids-codes-
2) Problems 1-4
area
Exs. 16-21 all, 22, 25, 26-36
Kuta software for
worksheet:
even
MP.1
www.kutasoftware.com/.
../10Surface%20Area%20of
%20Pyramids%20a..
MP.3
you tube video tutorial:
MP.6
www.youtube.com/watch
4.
Average:
1) problems 1-4
Exs. 9-15 odd, 44-53.
2) Problems 1-4
Exs. 17-21 odd, 22-38
MP.7
5.
Advanced:
1) Problems 1-4
Exs. 9-21 odd, 21-53
Math
1. Find the lateral area of
the entire cone shown
above.
power point notes:
2. Find the lateral area of
the small cone that has a
radius of 3 inches and a
www.marianhs.org/.../12.3%20Surface%20
Area%20of%20Pyramids
Google search:
102
height of 4 inches.
Use your answers to
E rci “a” an “ ”
find the amount of material
needed to make the
Elizabethan collar shown.
Math skills practice:
www.ixl.com/math/grade-8/surface-area-ofpyramids-and-cones
103
Apply trigonometric G.SRT.8
Pearson
Kuta software for
Standardized Test Prep (SAT/HSPA)
ratios:
worksheets
Use trigonometric ratios Chapter #11
Text book page 715 Q. 44-48
HSPA
and the Pythagorean
Get Ready www.kutasoftware.co
m/.../10PREP/PARCC/S
Theorem to solve right
Surface%20Area%20o
triangles in applied
AT
f%20Pyramids
problems.
Basic skills
Review:
30 mn
The figure above
represents a plan for a
wheelchair ramp to a
step that has a height of
10 inches. Jodi and
Kevin each used righttriangle trigonometry to
determine the length of
the ramp. Both solutions
are shown below.
Explain why both
solutions resulted in the
same answer.
Text book
page # 715
104
105
11.4
Volume of
prisms and
cylinders
Find the volume of a G.GMD.1
prism and the volume
Give an informal
of a cylinder.
argument for the
Example:
formulas for the
circumference of a
a. How do the radius and
circle, area of a circle,
height of the mug compare
volume of a cylinder,
to the radius and height of
pyramid, and cone.
the dog bowl?
Use dissection
b) How many times greater arguments, Cavalieri’s
is the volume of the bowl principle, and informal
limit arguments.
than the volume of the
mug?
2 days
Pearson
Chapter
#11.4 Get
Ready
Web link for finding 6.
the volume of a prism
and the volume of a
cylinder.
hotmath.com/help/gt/gen
ericprealg/section_9_6.ht
Kuta software for
worksheet:
Exs 6-13 all, 46-53
2) Problems 3-4
7.
www.kutasoftware.com/...
/10Volume%20of%20Prism
s%20and%20Cyli
you tube video tutorial:
MP.3
www.khanacademy.org/m
ath/.../volume.../solidgeometry-volu
MP.7
1) problems 1-2
Exs. 14-21 all, 24, 30- 32 all, 38
Text book
page #717724
MP.1
MP.6
Basic:
Average:
1) problems 1-2
Exs. 7-13 odd, 46-53.
2) Problems 3-4
Exs. 15-19 odd, 21-42
8.
Advanced:
1) Problems 1-4
Exs. 17-19 odd, 21-53
Math
power point notes:
Google search:
www.lms.stjohns.k12.fl.us/.../8th%20Std%20
Chapter%209%20PowerPoi
Math skills practice:
106
www.ixl.com/math/grade-8/volume-ofprisms-and-cylinders
107
Basic skills
Review:
Use trigonometric
ratios.
G.SRT.8
Pearson
Use trigonometric ratios Chapter #11
Example:
HSPA
and the Pythagorean
Get Ready
PREP/PARCC/S
Theorem to solve right
A 12-foot ladder that is
triangles in applied
AT
leaning against a wall
problems.
makes a 75.50
30 mn
Angle with the level
ground. Which of the
equations below can be
used to determine the
height, y, above the
ground, in feet, that the
ladder touches the wall?
(Sketch the diagram)
Kuta software for
worksheets
Standardized Test Prep (SAT/HSPA)
Text book page 724 Q. 46-49
www.kutasoftware.com/.../10Volume%20of%20Prisms%20a
nd%20Cyli
Text book
page # 724
108
109
Find the volume of a
pyramid and the
Volumes of volume of a cone.
pyramids and
Example:
cones
11.5
Popcorn A movie theater
G.GMD.3
Use volume formulas Chapter
for cylinders,
#11.5 Get
pyramids, cones, and Ready
spheres to solve
problems.
serves a small size of
popcorn in a conical
G.MG.1
container and a large size of
Use geometric
popcorn in a cylindrical
container.
shapes, their
measures, and their
properties to describe
objects (e.g., modeling
a tree trunk or a
human torso as a
cylinder).
2 days
a) What is the volume of the
small container? What is the MP.1
volume of the large
MP.3
container?
MP.7
b) How many small
containers of popcorn do you
have to buy to equal the
amount of popcorn in a large
container?
c) Which container gives you
more popcorn for your
money? Explain your
Pearson
Text book
page #726732
Web link for finding 9.
the volume of a
pyramid and cone:
Basic:
1) problems 1-2
Exs 5-14 all, 39-46
www.glencoe.com/sec/m
ath/prealg/mathnet/pr01/p
2) Problems 3-4
Exs. 15-21 all, 24-32, even
Kuta software for
worksheet:
www.mybookezzz.com/k 10.
uta-software-volume-ofpyramids-and-cones
Average:
1) problems 1-2
you tube video tutorial:
Exs. 5-13 odd, 39-46.
www.youtube.com/watch
2) Problems 3-4
Exs. 15-19 odd, 20-34
11.
Advanced:
1) Problems 1-4
Exs. 5-19 odd, 20-46
Math
power point notes:
Google search:
jcs.k12.oh.us/.../PH_Geo_115_Volumes_of_Pyramids_and_Cones.ppt
Math skills practice:
110
reasoning
hotmath.com/help/gt/genericprealg/section_
111
Basic skills
Review:
Use the Pythagorean
Theorem:
Example:
HSPA
PREP/PARCC/S
AT
G.SRT.8
Pearson
Use trigonometric ratios Chapter #11
and the Pythagorean
Get Ready
Theorem to solve right
triangles in applied
problems.
Kuta software for
worksheets
Standardized Test Prep (SAT/HSPA)
Text book page 732 Q. 39-42
www.kutasoftware.com/...
/10Volume%20of%20Pyram
ids%20and%20C..
Text book
page # 732
30 mn
The figure above
represents a plot of land
that Susan has
measured to use as a
fenced garden. Before
she builds the fence,
she wants to make sure
that the plot is
rectangular. She
measures the length of
one of the diagonals. If
Susan's plot is
rectangular, what will be
the length, in feet, of the
diagonal she
measured?
112
113
Find the surface area G.GMD.3
Pearson
and volume of a
Use volume formulas Chapter
Surface areas sphere.
for cylinders,
and volume of
#11.6 Get
Example:
pyramids, cones, and Ready
spheres
spheres to solve
The entrance to the Civil problems.
Rights Institute in
Birmingham, Alabama, G.MG.1
Text book
includes a hemisphere
page #733that has a radius of 25.3 Use geometric
740
shapes, their
feet.
measures, and their
properties to describe
objects (e.g., modeling
a tree trunk or a
2 days
human torso as a
cylinder).
11.6
a) Find the volume of the
hemisphere.
MP.1
b) Find the surface area
of the hemisphere, not
MP.3
including its base.
MP.6
c) The walls of the
hemisphere are 1.3 feet MP.7
thick. So, the rounded
surface inside the
building is a
hemisphere with a radius
of 24 feet. Find its
surface area, not
Web link for finding 12.
the surface area and
volume of a sphere.
Basic:
1) problems 1-2
Exs 6-16 all, 60-71
www.murrieta.k12.ca.us/c
ms/lib5/CA01000508/Cen
tricity/.../T9.6
2) Problems 3-4
Exs. 17-26 all, 29-31, 34-42
Kuta software for
worksheet:
Even, 50
www.kutasoftware.com/Fr 13.
eeWorksheets/GeoWork
Average:
1) problems 1-2
sheets/10-Spheres.pdf
Exs. 7-15 odd, 60-71.
you tube video tutorial:
2) Problems 3-4
www.khanacademy.org/m
ath/.../volume.../v/volum
e-of-a-sphere
Exs. 17-25 odd, 26-54
14.
Advanced:
1) Problems 1-4
Exs. 7-25 odd, 26-71
Math
power point notes:
Google search:
www.dgelman.com/powerpoints/.../12.6%2
0Surface%20Area
114
including its base.
Math skills practice:
www.ixl.com/math/grade-8/volume-andsurface-area-of-spheres
115
Basic skills
Review:
Use the inverse of
trigonometric ratios.
Example:
HSPA
PREP/PARCC/S
AT
The diagram below
shows a model of a
staircase in which all the
riser heights are equal
and all the tread lengths
are equal.
G.SRT.8
Pearson
Use trigonometric ratios Chapter #11
and the Pythagorean
Get Ready
Theorem to solve right
triangles in applied
problems.
Kuta software for
worksheets
Standardized Test Prep (SAT/HSPA)
Text book page 732 Q. 39-42
www.kutasoftware.com/FreeWo
rksheets/GeoWorksheets/10Spheres.pdf
Text book
page # 740
30 mn
A carpenter wants to
build a staircase in
which each riser has a
height of 6 inches and
each tread has a length
of 11 inches. Which of
the following
expressions is equal to
the
stair
angle?
116
117
Compare and find the G.MG.1
Pearson
areas and volumes of
Use geometric
Areas and
Chapter
similar solids.
shapes, their
Volumes of
#11.7 Get
Example:
measures, and their Ready
Similar Solids
properties to describe
Spheres in Architecture In
objects (e.g., modeling
Exercises a–d, refer to a tree trunk or a
the information below
Text book
human torso as a
about The Rose Center cylinder).
page #742for Earth and Space at
749
New York City’s
American Museum of
G.MG.2
Natural History. The
sphere has a diameter of
Apply concepts of
87 feet. The glass cube
density based on area
2 days
surrounding the sphere
and volume in
is 95 feet long on each
modeling situations
edge.
(e.g., persons per
square mile, BTUs per
cubic foot).
11.7
MP.3
15.
Web link for
comparing and
finding the areas and
volumes of similar
solids.
Basic:
1) problems 1-2
Exs 5-14 all, 42-54
2) Problems 3-4
www.ck12.org/geometry/
Area-and-Volume-ofSimilar-Solids
Exs. 15-26 all, 28-29, 34-38
Even
16.
Kuta software for
worksheet:
Average:
1) problems 1-2
Exs. 5-13 odd, 42-54.
www.kutasoftware.com/Fr
eeWorksheets/.../10Similar%20Solids
2) Problems 3-4
Exs. 15-23 odd, 24-38
you tube video tutorial:
17.
www.youtube.com/watch
-
Advanced:
1) Problems 1-4
Exs. 5-23 odd, 24-54
Math
power point notes:
MP.7
Google search:
MP.8
a) Find the surface area
of the sphere.
cs.k12.oh.us/.../PH_Geo_117_Areas_and_Volumes_of_Similar_Solid
s
118
b) Find the volume of
the sphere.
c) Find the volume of
the glass cube.
Math skills practice:
www.ck12.org/geometry/Area-and-Volumeof-Similar-Solids
d) Find the approximate
amount of glass used to
make the cube. (Hint:
Do not include the
ground or roof in your
calculations)
119
Basic skills
Review:
Apply the formula to
find area of oblique
triangles.
HSPA
PREP/PARCC/S Example:
AT
G.SRT.9
Pearson
Derive and use the
Chapter #11
formula for the area of Get Ready
an oblique triangle (A =
1/2 ab sin (C)).
Kuta software for
worksheets
Standardized Test Prep (SAT/HSPA)
Text book page 749 Q. 42-46
www.kutasoftware.com/FreeWo
rksheets/.../10Similar%20Solids
Text book
page # 749
30 mn
The figure above shows
a triangle drawn over a
map of Honduras. Use
the measurements of
the triangle to
approximate the area, in
square kilometers, of
Honduras. Show your
work.
INSTRUCTIONAL FOCUS OF UNIT
Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area
and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the
result of rotating a two-dimensional object about a line.
120
PARCC FRAMEWORK/ASSESSMENT
Square and circles: http://balancedassessment.concord.org/hs012.html
Examples:
A cookie factory is making cookies in a pyramid shape with equilateral triangle as a base. We know that the lateral edge of the cookie is 2 cm long and the base edge of
the cookie is 3 cm long.
1.
Prove that the height of the cookie is 1 cm.
2.
Find the volume of the cookie.
3.
Each cookie is wrapped totally in an aluminum foil. Prove that the minimum surface of foil necessary to wrap 100 cookies is greater than 960 cm2.
Example 2:
Diana’s Christmas present is placed into a cubic shaped box. The box is wrapped in a golden paper.
a. Are 3 m of golden paper enough for wrapping?
2
b. If 1 m of golden paper cost $3, how much would the wrapping material cost?
2
Could you pour 1 liter of juice in the box? (Know that 1 l=0.001 m3)
Wiki page for Common Core Assessments:
http://maccss.ncdpi.wikispaces.net/file/view/Geometry%20Unpacking062512%20.pdf/359510147/Geometry%20Unpacking062512%20.pdf
PARCC Framework Assessment questions with Model Curriculum Website for all units:
121
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGeo_Nov2012V3_FINAL.pdf
http://www.nj.gov/education/assessment/hs/hspa_mathhb.pdf
21ST CENTURY SKILLS
(4Cs & CTE Standards)
4.
Career Technical Education (CTE) Standards
1.
21 Century Life and Career Skills: All students will demonstrate the creative, critical thinking, collaboration, the problem-solving skills needed to function successfully
as both global citizens and workers in diverse ethnic and organizational cultures.
2.
Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings,
investment, and charitable giving in the global economy.
3.
Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the process of career awareness, exploration, and
preparation in order to navigate the globally competitive work environment of the information age.
4.
Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging
and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees.
st
122
1.
Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for careers in
emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees.
9.4.12.B.4: Perform math operations, such as estimating and distributing materials and supplies, to complete
classroom/workplace tasks.
Project Base Learning Activities:
http://www.achieve.org/files/CCSS-CTE-Task-IvySmith-GrowsUp-FINAL.pdf
MODIFICATIONS/ACCOMMODATIONS
Group activity or individual activity
1.
Review and copy notes from eno board/power point/smart board etc.
2.
Group/individual activities that will enhance understanding.
3.
Provide students with interesting problems and activities that extend the concept of the lesson
4.
Help students develop specific problem solving skills and strategies by providing scaffolded guiding questions
123
Peer tutoring
1.
Team up stronger math skills with lower math skills
Use of manipulative
2.
Eno or smart boards
3.
Dry erase markers
4.
Reference sheets created by special needs teacher
5.
Pairs of students work together to make word cards for the chapter vocabulary
6.
Use 3D shapes for visual learning
7.
Reference sheets for classroom
8.
Graphing calculators
APPENDIX
124
(Teacher resource extensions)
9.
CCSS. Mathematical Practices:
MP1: Make
sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry
points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form
and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems,
and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress
and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the
viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences
between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for
regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem.
Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this
make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different
approaches.
MP2: Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two
complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and
represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their
referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the
symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units
involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations
125
and objects.
MP3: Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in
constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are
able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate
them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the
context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments,
distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can
construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct,
even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies.
Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve
the arguments.
MP4: Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the
workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply
proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a
design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply
what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision
later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables,
graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their
mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served
its purpose.
126
MP5: Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil
and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic
geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when
each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high
school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade
levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve
problems. They are able to use technological tools to explore and deepen their understanding of concepts.
MP6: Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and
in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are
careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately
and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give
carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of
definitions.
MP7: Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and
seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.
Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression
x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and
127
can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 –
3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
MP8: Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper
elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they
have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with
slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x –
1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to
solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the
reasonableness of their intermediate results.
UNIT 4
Connecting Algebra And Geometry Coordinates
Total Number of Days: 11 days Grade/Course: __Geometry 10th grade__
ESSENTIAL QUESTIONS
10.
How do you prove that two lines are parallel or
ENDURING UNDERSTANDINGS
14.
A line can be graphed and its equation written when certain facts about the line
128
11.
12.
13.
perpendicular?
such as the slope and a point on the line are known.
How do you write an equation of a line in the coordinate
15.
plane?
The equations of a line can be written in various forms such as the Slope-intercept
form and the Point-slope form.
16.
How can you classify quadrilaterals?
How can you use coordinate geometry to prove general
17.
relationship?
Comparing the slopes of two lines can show whether the lines are parallel,
perpendicular, or neither.
The relationship between parallel or perpendicular lines can sometimes be used to
write the equation of a line.
18.
The formulas for slope, distance and midpoint can be used to classify and to prove
geometric relationships for figures in the coordinate plane.
19.
Using variables to name the coordinates of a figure allows relationships to be
shown to be true for a general case
20.
Geometric relationships can be proven using variable coordinates for figures in the
coordinate plane.
129
PACING
CONTENT
SKILLS
STANDARDS
RESOURCES
LEARNING
ACTIVITIES/ASSESSMENTS
OTHER
Pearson
(e.g., tech)
Prepares for
Text book http://www.regentsprep
G.GPE.5 Prove
page 189- .org/Regents/math/geo
Example:
the slope criteria
metry/GCG1/EqLines.ht Page 193: 1-7
196
for parallel and
m
The slope of a line is the rate of change
perpendicular
and is represented by m
lines and
Basic – Problems 1-2
Graph and write linear equations
2 days
3.7 Equations
of Lines in the
Coordinate
Plane
When a line passes through the points
(x1, y1) and (x2, y2), the slope (m) is
uses 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).
http://www.mathwareh Ex. 8 – 29, 50 – 63, 69 -77
ouse.com/algebra/linear
Average – Problems 1 – 4
_equation/slope-of-a-21.
line.php
Ex. 9 – 29 odd, 50 –63, 669 - 77
22.
Advanced – Problems 1-4
Ex 9-29 odd, 50-63, 69 - 77
Equations of line can take on several
MP 1
forms:
Slope Intercept Form:
[used when you know, or can find, the
slope, m, and the y-intercept, b.]
y = mx + b
MP 3
Point Slope Form:
[used when you know, or can find, a
point on the line (x1, y1), and the
slope, m.]
130
y – y1 = m(x – x1)
Write the equation of a line
Basic Skills
Review
30 mins
PARCC/
HSPA PREP
G.GPE.5
See above
Text book
page 196
http://www.regentsprep.o Standardized Test Prep
rg/Regents/math/geometr
Q 64 – 68
y/GCG1/EqLines.htm
Question:
What is the equation of the line in
slope-intercept form of the line parallel
to y = 5x + 2 that passes through the
point with coordinates (-2, 1)
http://www.mathwarehou Worksheets from stated websites.
se.com/algebra/linear_equ
ation/slope-of-a-line.php
131
Relate slope to parallel and
perpendicular lines
Example:
2 days
G.GPE.5 Prove Text book
the slope
page 197criteria for
204
parallel and
If we look at both equations, we notice perpendicular
that they both have slopes of 2. Since lines and
both lines "rise" two units for every one
uses them to
unit they "run," they will never
intersect. Thus, they are parallel lines. solve geometric
The graph of these equations is shown problems (e.g.,
3.8
below.
find the
equation of a
Slopes of
line
Parallel and
25.
http://www.regentsprep
26.
.org/Regents/math/ALG
EBRA/AC3/Lparallel.htm
Page 201: 1-6
Basic – Problems 1-2:
Ex. 7 – 14, 27, 53 - 62
Average – Problems 1 – 2:
Ex. 7 - 13 odd, 27, 53 - 62
Advanced – Problems 1-2:
Ex 7-21 odd, 27, 53 - 62
Kuta Software for
worksheets.
Perpendicular
Lines
http://www.wyzant.com
23.
/help/math/geometry/li
24.
nes_and_angles/parallel
_and_perpendicular
parallel or
perpendicular to
a given line that
passes through
a given
point).
MP 1
MP2
30 mins
Basic Skills Classify lines as parallel, perpendicular G.GPE.5
or neither.
Review
See above
Question:
PARCC/
Text book
page 204
http://www.regentsprep Standardized Test Prep
.org/Regents/math/geo
metry/GCG1/EqLines.ht Q 48 - 52
m
HSPA PREP Classify each of the following pairs of
lines as parallel, perpendicular or
132
neither
Parallel/
http://www.mathwareh Worksheets from stated websites.
ouse.com/algebra/linear
_equation/slope-of-a-27.
line.php
Lines
Perpendicular /Neither
3y= -5x -5
(y – 7) = 0.6(x – 5)
2x + 3y = 4
4x + 5y = 6
y = 4x + 1
(y – 2) = 4(x – 3)
y = -3x + 5
9x + 3y = 2
133
134
6.7
1 day
Polygons in
the
Coordinate
Plane
Classify polygons in the coordinate
Text book Teacher made power28.
G.GPE.7
plane applying the formulas for slope,
page 400 – point presentations
Use
coordinates
29.
distance and midpoint.
405
to compute
perimeters of
Example:
polygons and
30.
Is parallelogram WXYZ a rhombus?
areas of
triangles and
rectangles, e.g.,
31.
Explain
using the
distance
formula.
Page 403: 1-4
Basic – Problems 1-3,
Ex. 5– 18, 21 - 24, 31, 35 – 44,
49 - 54
Average – Problems 1 – 3,
Ex. 5 - 15 odd, 17 – 44, 49 - 54
Advanced – Problems 1-3,
Ex 5-15 odd, 17 - 44, 49 - 54
MP 1
MP 3
MP 8
135
Determine if a given polygon is a
triangle, parallelogram or a
quadrilateral
G.GPE.5
G.GPE 7
Use coordinates
to compute
perimeters of
Question:
polygons and
In the coordinate plane, quadrilateral areas of
ABCD has vertices with
triangles and
coordinates
rectangles, e.g.,
A(1, -1), B(-5, 3), C(-3, 6), and using the
Basic Skills
D(3, 2).
distance
Review
1) Compute the lengths of the sides of formula.
30 mins
PARCC/
HSPA PREP
Text book
page 405
Teacher made power
point presentations
Kuta software for
worksheets
Standardized Test Prep
Q 45 - 48
Worksheets from stated websites.
quadrilateral ABCD.
AB = ____
BC = ___
CD = ____
DA = ___
2) Compute the slopes of the sides
AB and AD .
Slope of AB = _____
Slope of AD = _____
136
3. Indicate in the table below whether
ABCD is an example of each shape
listed. Explain why it is or is not.
Shape
Yes or
Explain
No
Parallelogram
Rhombus
Rectangle
Square
137
Name coordinates of special figures by Prepares for
using their properties
G.GPE.4
Question
6.8
2 days
30 mins
SQRE is a square where SQ = 2a. The
axes bisect each side, what are the
coordinates of the vertices of SQRE?
Teacher made Power32.
point presentations
33.
Use coordinates to
Text book
prove simple
geometric theorems page 402 algebraically. For
412
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 the point
(0, 2).
Applying
Coordinate
Geometry
Find the coordinates of vertices of a G.GPE.4
Basic Skills polygon given coordinates of two
Review
vertices and a point of intersection of See above
the diagonals.
PARCC/
Kuta software
worksheets.
Page 403: 1-3
Basic – Problems 1-3,
Ex. 7– 13, 14, 17, 19, 23, 24, 28,
42-49
34.
Average – Problems 1 – 3,
Ex. 7- 13 odd, 14 – 31, 42 - 49
35.
http://www.mathopenr
ef.com/coordsquare.ht
ml
Advanced – Problems 1-3,
Ex 7-13 odd, 14 - 41, 42 - 49
http://www.youtube.co
m/watch?v=EZtXevirdes
Text book
page 412
http://www.mathopenr Standardized Test Prep
ef.com/coordsquare.ht
Q 38 - 41
ml
Worksheets from stated websites.
HSPA PREP
Question:
A parallelogram has two vertices at (1,
http://www.youtube.co
m/watch?v=EZtXevirdes
138
1) and (0, 7) and its diagonals cross at
the point (4, 3). Where are the other
two vertices of the parallelogram
139
Prove theorems using figures in the
coordinate plane.
G.GPE.4
Use coordinates
to prove simple Text book
geometric
page 414theorems
418
algebraically.
For example,
prove or
disprove that a
figure defined
6.9
2 days
Proofs Using
Coordinate
Geometry
Example:
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 the
point (0, 2).
MP 1
MP 3
MP 7
http://www.youtube.com
36.
/watch?v=EZtXevirdes
Page 416: 1-3
37.
Basic – Problems 1-2
Ex. 4, 6– 20 even, 21, 23, 33 40
38.
http://on.aol.com/video/
how-to-write-coordinateproofs-516909807
Average – Problems 1 – 2,
Ex. 4 - 14, 15 – 26, 33 – 40.
39.
Advanced – Problems 1-2,
Ex 4 – 28, 33 - 40
http://www.regentsprep.
org/Regents/math/geome
try/GCG4/CoordinatepRA
CTICE.htm
http://hotmath.com/hot
math_help/topics/coordin
ate-proofs.html
http://www.whiteplainsp
ublicschools.org/cms/lib5/
NY01000029/Centricity/D
omain/360/Coordinate%2
0Geometry%20Proofs%20
Packet%202012.pdf
140
Use Coordinate Geometry to Prove
Right Triangles and Parallelograms
Basic Skills
Review
Question:
30 mins
PARCC/
Daniel and Isaiah see a drawing of
quadrilateral ABCD, A(2,2), B(5,-2),
HSPA PREP
C(9,1) and D(6,5).
Daniel says the figure is a rhombus, but
not a square. Isaiah says the figure is a
square. Write a proof to show who is
making the correct observation.
G.G.PE 4
See above
Text book
page 418
http://www.regentsprep Standardized Test Prep
.org/Regents/math/geo
metry/GCG4/Coordinate Q 29 - 32
pRACTICE.htm
Worksheets from stated websites.
http://www.whiteplains
publicschools.org/cms/li
b5/NY01000029/Centrici
ty/Domain/360/Coordin
ate%20Geometry%20Pr
oofs%20Packet%202012
.pdf
141
To write the equation of a circle and
find the center and radius of a circle.
Example:
Definition: A circle is a locus (set) of
points in a plane equidistant from a
fixed point.
Circle whose center is at the origin
12.5
1 day
Equation:
Circles in the
Coordinate
plane
Example: Circle with center (0,0),
radius 4
G.GPE 1 Derive Text book
the equation of page 798 a circle of given 803
center and
radius using the
Pythagorean
Theorem;
complete the
square to find
the center and
radius of a circle
given by an
equation.
http://www.regentsprep
40.
.org/Regents/math/algtr
41.
ig/ATC1/circlelesson.ht
m
42.
http://www.mathwareh
ouse.com/geometry/circ
43.
le/equation-of-acircle.php
Page 800: 1-7
Basic – Problems 1-3
Ex. 8 – 30 all, 31 – 52 even, 58 65
Average – Problems 1 – 3,
Ex. 9 – 29 odd, 315– 56, 58 –
65.
Advanced – Problems 1-3,
Ex 9 – 29 odd, 31 - 65
http://www.mathsisfun.
com/algebra/circleequations.html
http://www.mathopenr
ef.com/coordgeneralcirc
le.html
142
Graph:
Circle whose center is at (h, k)
( i wi
r f rr
ra iu f r ”
I ay a
f r ”)
r f rr
a
“c n r-
a “ an ar
Equation:
143
Example: Circle with center (2,-5),
radius 3
Graph:
144
G.GPE 2 Derive Text book http://www.mathsisfun. Concept Byte Page 804
the equation of page 804 – com/geometry/parabola
Example
Activity 1
a parabola given 805
.html
a focus and
Activity 2
12.5
Definition: A parabola is a curve where
directrix.
any point is at an equal distance from:
Circles in the
http://www.purplemath Activity 3
Coordinate
1.
a fixed point (the focus), and
.com/modules/parabola. Ex: 17 - 23
plane
2.
a fixed straight line
htm
To write the equation of a parabola.
1 day
(the directrix)
http://hotmath.com/hot
math_help/topics/findin
g-the-equation-of-a145
parabola-given-focusand-directrix.html
* the axis of symmetry (goes through
the focus, at right angles to the
directrix)
* the vertex (where the parabola
makes its sharpest turn) is halfway
between the focus and directrix.
146
G.GPE 1 Derive Text book
the equation of page 803
Question:
a circle of given
In the coordinate plane, the circle center and
radius using the
with radius r centered at  h, k 
Pythagorean
consists of all the points  x, y  that Theorem;
are r units from  h, k  . Use the complete the
square to find
Basic Skills
Pythagorean theorem and the figure
the center and
Review
below to find an equation of the circle
radius of a circle
with
radius
r
and
center
PARCC/
given by an
equation.
Explain your answer.
HSPA PREP
Find the equation of a circle
30 mins
http://www.mathwareh Standardized Test Prep
ouse.com/geometry/circ
Q 58 - 60
le/equation-of-acircle.php
Worksheets from stated websites.
http://www.mathsisfun.
com/algebra/circleequations.html
INSTRUCTIONAL FOCUS OF UNIT
3.
4.
Building on their knowledge and work with the Pythagorean theorem, students will find distances in the coordinate plane.
Students will use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and
slopes of parallel and perpendicular lines.
147
5.
Students will continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola.
PARCC FRAMEWORK/ASSESSMENT
PARCC EXEMPLARS www.parcconline.org
1.
The coordinates are for a quadrilateral, (3, 0), (1, 3), (-2, 1), and (0,-2). Determine the type of quadrilateral made by connecting these four points?
Identify the properties used to determine your classification. You must give confirming information about the polygon.
2.
If Quadrilateral ABCD is a rectangle, where A(1, 2), B(6, 0), C(10,10) and D(?, ?) is unknown. a. Find the coordinates of the fourth vertex. b. Verify that
ABCD is a rectangle providing evidence related to the sides and angles.
3.
Derive the equation of the parabola that has the focus (1, 4) and the directrix x=-5.
4.
Derive the equation of the parabola that has the focus (2, 1) and the directrix y=-4.
5.
Derive the equation of the parabola that has the focus (-3, -2) and the vertex (1, -2).
6.
Given the midpoint of a segment and one endpoint. Find the other endpoint.
a. Midpoint: (6, 2) endpoint: (1, 3)
b. Midpoint: (-1, -2) endpoint: (3.5, -7)
148
7. Investigate the slopes of each of the sides of the rectangle ABCD (shown below). What do you notice about the slopes of the sides that meet at a right
angle? What do you notice about the slopes of the opposite sides that are parallel? Can you generalize what happens when you multiply slopes of
perperpendicular lines?
8. If general points N at (a,b) and P at (c,d) are given. Why are the coordinates of point Q (a,d)? Can you find the coordinates of point M?
149
9. Jennifer and Jane are best friends. They placed a map of their town on a coordinate grid and found the point at w ic
u i a ( , ) an Jan ’
u i a ( , ) an
y wan
in
i
, w a ar
c r ina
f
ac f
ir u i If J nnif r’
ace they should meet?
10. John was visiting three cities that lie on a coordinate grid at (-4, 5), (4, 5), and (-3, -4). If he visited all the cities and ended up where he started, what is the
distance in miles he traveled?
11. Suppose a line k in a coordinate plane has slope c/d
a. What is the slope of a line parallel to k? Why must this be the case?
b. What is the slope of a line perpendicular to k? Why does this seem reasonable?
12. Two points A(0, -4) , B(2, -1) determines a line, AB.
a. What is the equation of the line AB?
b. What is the equation of the line perpendicular to AB passing through the point (2,-1)?
150
Wiki page for Common Core Assessments
http://maccss.ncdpi.wikispaces.net/file/view/Geometry%20Unpacking062512%20.pdf/359510147/Geometry%20Unpacking062512%20.pdf
PARCC Framework Assessment questions with Model Curriculum Website for all units
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGeo_Nov2012V3_FINAL.pdf
http://www.nj.gov/education/assessment/hs/hspa_mathhb.pdf
21ST CENTURY SKILLS
(4Cs & CTE Standards)
Career Technical Education (CTE) Standards
1.
21st Century Life and Career Skills: All students will demonstrate the creative, critical thinking, collaboration, the problem-solving skills needed to
function successfully as both global citizens and workers in diverse ethnic and organizational cultures.
9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems,
using multiple perspectives.
2.
Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial responsibility related to financial
151
planning, savings, investment, and charitable giving in the global economy.
9.2.12.B.1 Prioritize financial decisions by systematically considering alternatives and possible consequences.
3.
Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the process of career awareness,
exploration, and preparation in order to navigate the globally competitive work environment of the information age.
9.3.12.C.2 Characterize education and skills needed to achieve career goals, and take steps to prepare for postsecondary options, including making
course selections, preparing for and taking assessments, and participating in extra-curricular activities.
4.
Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for
careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees.
9.4.12.B.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and
opportunities.
career
Project Base Learning Activities:
http://www.achieve.org/files/CCSS-CTE-Task-Stairway-FINAL.pdf
MODIFICATIONS/ACCOMMODATIONS
Group activity or individual activity
- Review and copy notes from eno board/power point/smart board etc.
- Group/individual activities that will enhance understanding.
- Provide students with interesting problems and activities that extend the concept of the l
152
- Help students develop specific problem solving skills and strategies by providing scaffolding guiding questions
Peer tutoring
- Team up stronger math skills with lower math skills
Use of manipulative
- Eno or smart boards
- Dry erase markers
- Reference sheets created by special needs teacher
- Pairs of students work together to make word cards for the chapter vocabulary
- Use 3D shapes for visual learning
- Reference sheets for classroom
- Graphing calculators
APPENDIX
(Teacher resource extensions)
MP1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze
givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than
153
simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain
insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically
proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and
solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continua y a k
v ,“
i ak
n ”
y can un r an
a r ac
f
r
ving c
r
an i ntify correspondences between different approaches.
MP 2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on
problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed
during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing
and flexibly using different properties of operations and objects.
MP 3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They
make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.
They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there
is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and
actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to
determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask
useful questions to clarify or improve the arguments.
MP 4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early
grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan
a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe
154
how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a
practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those
relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether
the results make sense, possibly improving the model if it has not served its purpose.
MP 5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete
models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students
are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing
both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions
generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making
mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare
predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital
content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of
concepts
MP 6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own
reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying
units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical
answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each
other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
MP 7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the
same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the
14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for
155
solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single
objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize
that its value cannot be more than 5 for any real numbers x and y.
MP 8 Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might
notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying
attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract
the equation (y – 2)/(x– 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 +x + 1), and (x – 1)(x3 + x2 + x + 1) might
lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of
the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
5.
Kuta G 1: Kuta Software – Geometry (Free Worksheets)
6.
Teacher Edition: Geometry Common Core by Pearson
7.
Student Companion: Geometry Common Core by Pearson
8.
Practice and Problem Solving Workbook: Geometry Common Core by Pearson
9.
Teaching with TI Technology: Pearson Mathematics by Pearson
10.
Progress Monitoring Assessments: Geometry Common Core by Pearson
11.
http://www.jmap.org/JMAP_RESOURCES_BY_TOPIC.htm#Geo
12.
http://www.mathopenref.com/
13.
http://www.mathisfun.com/
14.
http://www.mathwarehouse.com/
156
15.
http://www.regentsprep.org/Regents/math/geometry/math-GEOMETRY.htm
*
http://www.cpm.org/pdfs/state_supplements
*
http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqgeomapr15.pdf
* http://illuminations.nctm.org
* http://www.state.nj.us/education/cccs/standards/9/
*http://www.whiteplainspublicschools.org/cms/lib5/NY01000029/Centricity/Domain/360/Coordinate%20Geometry%20Proofs%20Packet%202012.pdf
NOTE:
Standards alignment in accordance with Appendix A of Common Core State Standards and Pear n’ G
1 and 2
ry C
nC r
ac
r’ E i i n V u
Notes to teacher (not to be included in your final draft):
4 Cs
Three Part Objective
157
Creativity: projects
Behavior
Critical Thinking: Math Journal
Condition
Collaboration: Teams/Groups/Stations
Demonstration of Learning (DOL)
Communication – Powerpoints/Presentations
UNIT 5
Circles With and Without Coordinates
Total Number of Days: 11days Grade/Course: __Geometry 10th grade__
ESSENTIAL QUESTIONS
ENDURING UNDERSTANDINGS
16.
How do you solve problems that involve measurements20.
of
triangles?
Angle bisectors and segment bisectors can be used in triangles to determine
various angle and segment measures.
17.
How do you find the area of a polygon or find the
circumference and area of a circle?
ng
in the circle.
18.
How can you prove relationships between angles and arcs
22.
in a circle?
The ar a f ar
radius is known
19.
How do you find the equation of a circle in the coordinate
23.
plane?
Angles formed by intersecting lines have a special relationship to the arcs the
intersecting lines intercept
21.
f ar
f a circ ’ circu f r nc can
f a circ f r
f un
y ra ii an arc can
y r a ing i
f un w
n
an ang
circ ’
158
24.
The information in the equation of a circle allows the circle to be graphed. The
equation of a circle can be written if its center and radius are known.
RESOURCES
STANDARDS
PACING
CONTENT
SKILLS
(CCSS/MP)
OTHER
Pearson
LEARNING
ACTIVITIES/ASSESSMENTS
(e.g., tech)
1 day
5.3
Bisectors in
-To identify properties of
perpendicular bisectors and angles
G.C.3
Construct the
Text book www.pkwy.k12.mo.us/ho
page 300- mepage/ataylor1/file/2.2.
pdf
Page 300 - Concept Byte: Paper
159
Triangles
inscribed and
circumscribed
Example:
circles of a
Technology Use geometry software to triangle, and
prove
draw ABC. Construct the angle
properties of
bisector of BAC. Then find the
angles for a
midpoint of
. Drag any of the points.
quadrilateral
Does the angle bisector always pass
inscribed in a
through the midpoint of the opposite
circle.
side? Does it ever pass through the
midpoint?
bisectors.
307
Folding Bisectors
http://www.jmap.org/Static
Files/PDFFILES/Workshee
tsByTopic/ANGLES/Drills Page 304: 1 - 6
/PR_Measuring_Angles_3.
pdf
Basic – Problems 1-3
Ex. 7– 20, 23, 26 – 29, 33 - 40
25.
Average – Problems 1 – 3
Ex. 7 – 17 odd, 18–29, 33 - 40
26.
Advanced – Problems 1-3
Ex 7-17 odd, 18-40.
160
30 mins
To write an argument for the formulas G.GMD.1
Text book
for the volume of a pyramid.
page 307
Give an informal
argument for
the formulas for
Basic Skills
Question:
the
Review
circumference
A cube in the xyz-coordinate system
PARCC/HSPA
(not shown) centered at the origin has of a circle, area
PREP
of a circle,
vertices at the points
volume of a
cylinder,
pyramid, and
cone. Use
dissection
and
arguments,
www.pkwy.k12.mo.us/ho Standardized Test Prep
mepage/ataylor1/file/2.2.
Q 33 – 36
pdf
http://www.jmap.org/Static Worksheets from stated websites.
Files/PDFFILES/Workshee
tsByTopic/ANGLES/Drills
/PR_Measuring_Angles_3.
pdf
161
, where
. If lines
are drawn from the center of the cube
to the 8 vertices of the cube, 6
pyramids are formed. Explain how
a pyramid with height a and square
Cava i ri’
principle, and
informal limit
arguments.
MP 1
3
base of side length 2a has a volume a
1 day
MP 3
To find the measures of central angles G.CO.1, G.C.1, Text book
and arcs, find the circumference and G.C.2
page 649arc length
658
Identify and
describe
Example:
relationships
Challenge Engineers reduced the lean among inscribed
of the Leaning Tower of Pisa. If they
angles, radii,
moved it back 0.46�, what was the arc
and chords.
length of the move? Round your
Include the
10.6
answer to the nearest whole number. relationship
between
Circles and
central,
Arcs
inscribed, and
circumscribed
angles; inscribed
angles on a
diameter are
right angles; the
radius of a circle
is perpendicular
to the tangent
where the
http://www.regentsprep Page 658 - Concept Byte: Circle
.org/Regents/math/geo Graphs
metry/GP15/CircleArcs.
htm
27.
Page 654: 1-8
28.
www.cpm.org/pdfs/skillB
uilders/GC/GC_Extra_Pr
actice_Section18.pdf
Basic – Problems 1-4:
Ex. 9 – 35, 36 – 50 even, 64 - 71
29.
Average – Problems 1 – 4:
Ex. 9- 35 odd, 36 – 56, 64 - 71
www.robertfant.com/Geo
30.
metry/PowerPoint/Chapt
er11.ppt
Advanced – Problems 1-4:
Ex 9-35 odd, 36 – 56, 64 - 71
162
radius intersects
the circle.
MP 1
MP 3
MP 8
30 mins
To find the measure of an angle in a G.C.2
circle
See above
Basic Skills Question:
Review
In the figure below,
is tangent to
PARCC/HSPA the circle with center O at point A. If
PREP
has a measure of 68 degrees,
what is the measure, in degrees,
Text book
page 657
163.150.89.242/yhs/F Standardized Test Prep
aculty/AB/bagg/Geom
etry/images/.../11.3.pd Q 60 - 63
f
Worksheets from stated websites.
http://www.mathopenr
31.
ef.com/arccentralanglet
heorem.html
of
163
164
10.7
1 day
Areas of
Circles and
Sectors
To find the areas of circles, sectors and G.C.5 Derive
Text book Teacher made power
using similarity page 659 – point presentations
segments of circles.
the fact that the 667
Example:
length of the arc
intercepted by
Landscaping The diagram shows the
an angle is
area of a lawn covered by a water
proportional to
sprinkler. Round your answer to the
the radius, and
nearest whole number.
define the
32.
1. What is the area of the lawn that is radian measure
covered by the sprinkler?
of the angle as
33.
the constant of
2. Suppose the water pressure is
proportionality;
weakened so that the radius is 12 feet.
derive the
What is the area of lawn that will be
34.
formula for the
covered?
area of a sector.
Page 659 - Concept Byte: Exploring
the Area of a Circle
35.
Advanced – Problems 1-3:
Ex 7-25 odd, 26 – 50, 55 - 63
MP 1
Page 667 – Concept Byte: Inscribed
and Circumscribed Figures
Page 663: 1-6
Basic – Problems 1-3:
Ex. 7 – 25, 26 – 34 even, 35 –
36, 55 - 63
Average – Problems 1 – 3:
Ex. 7 - 25 odd, 26 – 44, 55 - 63
MP 3
MP 6
MP 8
165
166
To find the length of a chord in a circle G.C.5
Question:
See above
Basic Skills
Review
30 mins
Text book
page 666
http://www.mathopenr Standardized Test Prep
ef.com/chord.html
Q 51 - 54
http://www.regentsprep
.org/Regents/math/geo Worksheets from stated websites.
metry/GP14/CircleSegm
ents.htm
A circle with center O and radius 5 has
PARCC/HSPA central angle XOY. If mXY = 600, what is
PREP
the length of chord XY?
A circle w
To use properties of a tangent to a
circle
2 days
G.C.2
Identify and
describe
Example:
relationships
You are standing at C, 8 feet from a silo. among inscribed
The distance to a point of tangency is angles, radii,
12.1
16 feet. What is the radius of the silo? and chords.
Include the
Tangent Lines
relationship
between
central,
inscribed, and
circumscribed
angles; inscribed
Text book Teacher made power
page 762 – point presentations
36.
769
37.
http://www.murrieta.k1
2.ca.us/cms/lib5/CA010
38.
00508/Centricity/Domai
n/1830/T11.2.pdf
39.
Page 766: 1-5
Basic – Problems 1-5:
Ex. 6 – 22, 26 – 31, 36 – 44,
Average – Problems 1 – 5:
Ex. 7 - 19 odd, 20 – 31, 36 - 44
Advanced – Problems 1-5:
Ex 7-19 odd, 20 – 31, 36 - 44
http://jmap.org/htmlsta
ndard/Geometry/Inform
al_and_Formal_Proofs/
G.G.50.htm
167
angles on a
diameter are
right angles; the
radius of a circle
is perpendicular
to the tangent
where the
radius intersects
the circle.
MP 1
MP 3
30 mins
To visualize the relation between two- G.GMD.4
Text book
dimensional and three-dimensional
page 769
Identify the
objects.
shapes of twodimensional
Basic Skills Question:
cross-sections of
Review
A three dimensional object is created threePARCC/HSPA by rotating a circle about one of its
dimensional
diameters. What is the shape of the
PREP
objects, and
resulting object? Give as much detail as identify threepossible.
dimensional
objects
generated by
rotations of
two-
Teacher made power Standardized Test Prep
point presentation.
Q 32 - 35
168
dimensional
objects.
A circle
To use congruent chords, arcs and
central angles and also apply
perpendicular bisectors to chords.
Example:
12.2
2 days
Chords And
Arcs
Find the Length of a Chord
G.C.2
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
Text Book
page 771 779
http://163.150.89.242/y Page 770 - Concept Byte: Paper
hs/Faculty/AB/bagg/Geo Folding With Circles
metry/images/Geometr
y%20text%20PDFs/11.4.
pdf
40.
Page 776: 1-5
41.
http://www.jmap.org/ht
mlstandard/Geometry/I
nformal_and_Formal_Pr
42.
oofs/G.G.52.htm
43.
Teacher made power
point presentation
Basic – Problems 1-4:
Ex. 6– 16, 18, 23 – 25, 29, 44 52
Average – Problems 1 – 4:
Ex. 7 - 15 odd, 16 – 34, 44 - 52
Advanced – Problems 1-4:
Ex 7-15 odd, 16 – 39, 44 - 52
169
right angles; the
radius of a circle
is perpendicular
to the tangent
where the
radius intersects
the circle.
MP 1
MP 3
To use the properties of tangent to
construct a line tangent to a circle.
30 mins
G.C.4 Construct Text book
a tangent line page 779
from a point
Question:
outside a given
Basic Skills
Construct a line through point P that is circle to the
Review
tangent to circle O below. Leave all circle.
PARCC/HSPA construction marks.
PREP
Construct a line through point.
Teacher made power Standardized Test Prep
point presentation.
Q 40 - 43
http://www.mathopenr
ef.com/consttangent.ht
ml
http://mathbits.com/Ma
thBits/GSP/TangentCircl
e.htm
170
http://www.youtube.co
m/watch?v=IT52gEoGe9
A
A circle
12.3
2 days
Inscribed
Angles
To find the measures of an inscribed G.C.2, G.C.3,
angle, measure of an angle formed by
Construct the
a tangent and a chord.
inscribed and
circumscribed
circles of a
Example:
triangle, and
Find the measure of the inscribed angle prove
properties of
or the intercepted arc.
angles for a
quadrilateral
inscribed in a
circle.
Text book
page 780 787
http://163.150.89.242/y
hs/Faculty/AB/bagg/Geo
44.
metry/images/Geometr
y%20text%20PDFs/11.5.
45.
pdf
Page 784: 1-5
Basic – Problems 1-3:
Ex. 6– 19, 18, 20 – 24 even, 28 29, 44 - 51
Teacher made power46.
point presentations.
Average – Problems 1 – 3:
Ex. 7 - 17odd, 19 – 34, 44 - 51
47.
Advanced – Problems 1-3:
Ex 7-17odd, 19 - 39, 44 - 51
http://www.jmap.org/ht
mlstandard/Geometry/I
nformal_and_Formal_Pr
171
MP 1
oofs/G.G.51.htm
MP 3
http://www.youtube.co
m/watch?v=DjgPtK0_Qh
0
http://www.mathopenr
ef.com/circleinscribed.h
tml
30 mins
To find the measure of an inscribed
Basic Skills angle.
Review
Question:
PARCC/HSPA
Quadrilateral WXYZ is inscribed in a
PREP
circle. If
radians
G.C.2, G.C.3
Above
Text book
page 787
Teacher made power Standardized Test Prep
point presentation.
Q 40 - 43
http://www.mathopenr
ef.com/circleinscribed.h
tml
172
and mX 
4
radians, what are
5
the measures, in radians, of the other
two angles in the quadrilateral?
1 day
To find measures of angles formed by G.C.2
Text book
chords, secants, tangents, and also
page 789 Identify
and
find the lengths of segments
797
describe
associated with circles
relationships
among inscribed
Example:
angles, radii,
and chords.
Include the
12.4
relationship
Angle
between
Measures and
central,
Segment
inscribed, and
Lengths
circumscribed
angles; inscribed
angles on a
diameter are
right angles; the
radius of a circle
is perpendicular
to the tangent
where the
Teacher made power
point presentation
http://www.finneytown.
48.
org/Downloads/GETE12
49.
042.pdf
http://www.youtube.co
50.
m/watch?v=Ax33G6YdS
v0
51.
Page 789 - Concept Byte: Exploring
Chords and Secants
Page 794: 1-7
Basic – Problems 1-3:
Ex. 8– 20, 22 = 26 even, 27 – 31
odd, 48 – 55
Average – Problems 1 – 3:
Ex. 9 - 19 odd, 21 – 39, 48 - 55
Advanced – Problems 1-3:
Ex 9-19 odd, 21 – 43, 48 - 55
http://www.jmap.org/ht
mlstandard/Geometry/I
nformal_and_Formal_Pr
oofs/G.G.51.htm
173
radius intersects
the circle.
MP 1
MP 3
To find the measure of an angle.
Question:
30 mins
In the figure below, ABC is
Basic Skills circumscribed about the circle centered
Review
at O. If the measure of AOC is
PARCC/HSPA
radians, what is the measure, in
PREP
radians, of
G.C.2, Above
Text book
page 797
Teacher made power Standardized Test Prep
point presentation.
Q 44- 47
http://www.jmap.org/ht
mlstandard/Geometry/I
nformal_and_Formal_Pr
oofs/G.G.51.htm
174
1 day
To write the equation of a circle and to G.GPE.1 Derive Text book
find the center and radius of a circle. the equation of page 798 a circle of given 803
center and
radius using the
Example:
Pythagorean
Using the Center and a Point on a
Theorem;
Circle.
complete the
12.5
square to find
Write the standard equation of the
the center and
Circles in the
circle with center (1, -3) that passes
radius of a circle
Coordinate
through the point (2, 2).
given by an
Plane
equation.
http://www.finneytown.
org/Downloads/GETE
52.
12052.pdf
53.
http://www.jmap.org/ht
54.
mlstandard/Geometry/
Coordinate_Geometry/
G.G.71.htm
55.
Page 800: 1-7
Basic – Problems 1-3:
Ex. 8– 30, 31 - 52 even, 61 – 65
Average – Problems 1 – 3:
Ex. 9 - 29 odd, 31 – 56, 61 - 65
Advanced – Problems 1-3:
Ex 9-29 odd, 31 – 56, 61 - 65
http://www.ck12.org/g
eometry/Circles-in-theCoordinate-Plane/
MP 1
MP 3
MP 7
175
To find the equation of a circle
Basic Skills Question:
Review
30 mins
PARCC/HSPA
PREP
G.GPE 1 Derive Text book
the equation of page 803
a circle of given
center and
radius using the
Pythagorean
Theorem;
complete the
square to find
the center and
http://www.mathwareh Standardized Test Prep
ouse.com/geometry/circ
Q 58 - 60
le/equation-of-acircle.php
Worksheets from stated websites.
http://www.mathsisfun.
com/algebra/circleequations.html
176
radius of a circle
given by an
equation.
In the coordinate plane, the circle with
radius r centered at  h, k  consists
 x, y 
 h, k  .
of all the points
that are r
units
Use
from
the
Pythagorean theorem and the figure
below to find an equation of the circle
with radius r and center
Explain your answer.
177
INSTRUCTIONAL FOCUS OF UNIT
56.
In this unit, students will prove basic theorems about circles, with particular attention to perpendicularity and inscribed angles, in order to see
symmetry in circles and as an application of triangle congruence criteria.
57.
They will study relationships among segments on chords, secants, and tangents as an application of similarity.
58.
In the Cartesian coordinate system, students will use the distance formula to write the equation of a circle when given the radius and the coordinates
of its center. Given an equation of a circle, they will draw the graph in the coordinate plane, and apply techniques for solving quadratic equations to
determine intersections between lines and circles or parabolas and between two circles.
PARCC FRAMEWORK/ASSESSMENT
PARCC EXEMPLARS www.parcconline.org
1.
An archeologist dug up an edge piece of a circular plate. He wants to know what the original diameter of the plate was before it broke. However, the
piece of pottery does not display the center of the plate. How could he find the original dimensions?
2.
Jessica works at a daycare center and she is watching three rambunctious toddlers. One of the toddlers is in a crib at point A, another toddler is in her
high chair at point B and the third toddler is in a play-station at point C. Where can Jessica position herself so that she is equidistant from each of the
children? Construct an argument using concrete referents such as objects, drawing, diagrams and actions.
3.
Since all circles are similar, the ratio of the
will be the scale factor for any circle. Two students use different reasoning to find the length
of an arc with central angle measure of 45° in a circle with radius=3cm. Compare the effectiveness of these two plausible arguments:
178
Sv
ana ay : “I kn w
Since
ic a
a 36
0
= 2 radians
= , the equivalent measure of the length of the arc will be (2 * 3) or
ay : “36
0
= 2 radians therefore 1 =
radians. 45 =(
c ”
) radians. Therefore the measure of the arc will be 45 =(
Can y u u Sv ana’ r a ning fin an arc ng a cia
wi a c n ra ang f
º f a circ wi a ra iu f f
to find the arc length of a circle with a radius of 6m and a central angle of 120º? Which method do you prefer and why?
4.
) radians * 3cm =
Can y u u
c ”
ic a ’ r a ning
Given a coordinate and a distance from that coordinate develop a rule that shows the locus of points that is that given distance from the given point
(based on the Pythagorean theorem). If the coordinate of point H in the diagram below is (x,y) and the length of DH is 4 units. Can you write a rule that
represents the relationship of the x value, the y value and the radius? Why is this relationship true? As point H rotates around the circle, does this
relationship stay true?
5. In the diagram below, circle D translated 4 units to the right to create circle E. Why is the equation of this new circle (x − 4)2 + y2 = 42 . Why is the equation
for circle I x2 + (y − 4)2 = 42 . Using similar reasoning, could you right and equation for a circle with the center at
(-4, 0) and a radius of 4? Center of (0, -4) and radius of 4? What is the equation of a circle with center at (-8,11) and a radius of
5 ? Can you generalize this
equation for a circle with a center at (h,k) and a radius of r?
179
ion of a
a directrix.
Add limitations for course 2 and 3.
G-GPE.2 Given a focus and directrix, derive the equation of a parabola.
Parabola is defined as “the set of all points P in a plane equidistant from a fixed line and a fixed point in the plane.”
The fixed
lineCommon
is called Core
the directrix,
and the fixed point is called the focus.
Wiki
page for
Assessments
(Level II)
http://maccss.ncdpi.wikispaces.net/file/view/Geometry%20Unpacking062512%20.pdf/359510147/Geometry%20Unpacking062512%20.pdf
Ex. Derive the equation of the parabola that has the focus (1, 4) and the directrix x=-5.
Ex. Derive the equation of the parabola that has the focus (2, 1) and the directrix y=-4.
Ex. Derive the equation of the parabola that has the focus (-3, -2) and the vertex (1, -2).
PARCC Framework Assessment questions with Model Curriculum Website for all units
quations of http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf
G-GPE.3 Given the foci, derive the equation of an ellipse, noting that the sum of the distances from the foci to any
en the foci, fixed point on the ellipse is constant, identifying the major and minor axis.
or
m the foci
G-GPE.3 Given the foci, derive the equation of a hyperbola, noting that the absolute value of the differences of the
distances from the foci to a point on the hyperbola is constant, and identifying the vertices, center, transverse axis,
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGeo_Nov2012V3_FINAL.pdf
conjugate axis, and asymptotes.
http://www.nj.gov/education/assessment/hs/hspa_mathhb.pdf
21ST CENTURY SKILLS
180
(4Cs & CTE Standards)
Career Technical Education (CTE) Standards
5.
21st Century Life and Career Skills: All students will demonstrate the creative, critical thinking, collaboration, the problem-solving skills needed to
function successfully as both global citizens and workers in diverse ethnic and organizational cultures.
9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems,
using multiple perspectives.
6.
Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial responsibility related to financial
planning, savings, investment, and charitable giving in the global economy.
9.2.12.B.1 Prioritize financial decisions by systematically considering alternatives and possible consequences.
7.
Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the process of career awareness,
exploration, and preparation in order to navigate the globally competitive work environment of the information age.
9.3.12.C.2 Characterize education and skills needed to achieve career goals, and take steps to prepare for postsecondary options, including making
course selections, preparing for and taking assessments, and participating in extra-curricular activities.
8.
Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for
careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees.
9.4.12.B.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career
opportunities.
Project Base Learning Activities:
http://www.achieve.org/files/CCSS-CTE-Spread-of-Disease-FINAL.pdf
181
MODIFICATIONS/ACCOMMODATIONS
Group activity or individual activity
- Review and copy notes from eno board/power point/smart board etc.
- Group/individual activities that will enhance understanding.
- Provide students with interesting problems and activities that extend the concept of the l
- Help students develop specific problem solving skills and strategies by providing scaffolding guiding questions
Peer tutoring
- Team up stronger math skills with lower math skills
Use of manipulative
- Eno or smart boards
- Dry erase markers
- Reference sheets created by special needs teacher
- Pairs of students work together to make word cards for the chapter vocabulary
- Use 3D shapes for visual learning
- Reference sheets for classroom
- Graphing calculators
182
APPENDIX
(Teacher resource extensions)
MP1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze
givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain
insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically
proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and
solve a problem. Mathematically proficient students check their answers to problems using a different method, an
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MP 2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on
problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed
during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing
and flexibly using different properties of operations and objects.
MP 3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They
make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.
They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there
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is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and
actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to
determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask
useful questions to clarify or improve the arguments.
MP 4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early
grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan
a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe
how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a
practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those
relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether
the results make sense, possibly improving the model if it has not served its purpose.
MP 5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete
models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students
are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing
both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions
generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making
mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare
predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital
content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of
concepts
MP 6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own
reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying
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units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical
answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each
other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
MP 7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the
same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the
14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for
solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single
objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize
that its value cannot be more than 5 for any real numbers x and y.
MP 8 Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might
notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying
attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract
the equation (y – 2)/(x– 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 +x + 1), and (x – 1)(x3 + x2 + x + 1) might
lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of
the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
9.
Kuta G 1: Kuta Software – Geometry (Free Worksheets)
10.
Teacher Edition: Geometry Common Core by Pearson
11.
Student Companion: Geometry Common Core by Pearson
12.
Practice and Problem Solving Workbook: Geometry Common Core by Pearson
13.
Teaching with TI Technology: Pearson Mathematics by Pearson
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14.
Progress Monitoring Assessments: Geometry Common Core by Pearson
15.
http://www.jmap.org/JMAP_RESOURCES_BY_TOPIC.htm#Geo
16.
http://www.mathopenref.com/
17.
http://www.mathisfun.com/
18.
http://www.mathwarehouse.com/
19.
http://www.regentsprep.org/Regents/math/geometry/math-GEOMETRY.htm
*
http://www.cpm.org/pdfs/state_supplements
*
http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqgeomapr15.pdf
* http://illuminations.nctm.org
* http://www.state.nj.us/education/cccs/standards/9/
*http://www.whiteplainspublicschools.org/cms/lib5/NY01000029/Centricity/Domain/360/Coordinate%20Geometry%20Proofs%20Packet%202012.pdf
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