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Overview
Our class is a high school geometry
section A. We will be using the High
School Holt Geometry book by Holt,
Rinehart, and Winston printed in 2007.
We have decided to focus on the
Common Core State Standard (CCSS)
Congruence G-CO and cluster
Experiment with transformations in the
plane. Students in this Geometry class
will experiment with transforming
images in the coordinate plane. These
transformations will consist of
rotating, reflecting, and translating
original images (referred to as
preimages) to new images (referred
as images and marked with a prime).
However, before arriving at this
common core standard, students must
master a cluster of activities. These
activities are dependent of each other:
G-CO.1
(A). Understanding points,
lines, & planes. Students will
identify points, rays, opposite
rays, lines, segments, and planes.
Students will draw and label
points, lines, planes, and rays.
G-CO.1
Students will be introduced to basic
postulate, as these will be used later on
in the chapter.
G-CO.1
A) Benchmark assessment related to
CCSS
Given the following diagram, Identify
two opposite rays, a point on ray BC,
identify the intersection of plane N and
plane T, and a plane containing E, D, &
B.
G-CO.1
B). Angles
Students will name and classify angles.
Students will measure and construct angles
and angle bisectors using tools such as a
protractor and a ruler. Students will be
introduced to the following terms: angle,
vertex, interior and exterior angles, measure,
degree, acute, obtuse, right and straight
angles, congruent angles, and angle bisector.
Write the different ways to name the
angles in the diagram on the right.
Answer: RTQ, T, STR, 1, 2
Example of using protractor
I.
If OC corresponds with c and
OD corresponds with d,
mDOC = |d – c| or |c – d|.
II.
Classifying angles by their
degrees
III.
An angle bisector is a ray
that divides an angle into two
congruent angles. JK bisects
LJM; thus LJK  KJM
Given the figure on the right,
find mJKM
KM bisects JKL, mJKM = (4x + 6)°,
and mMKL = (7x – 12)°.
Step 1find x
mJKM = mMKL
(4x + 6)° = (7x – 12)°
+12
+12
4x + 18 = 7x
-4x
= -4x
18 = 3x
6= x
Step 2 Find mJKM
mJKM = 4x + 6
= 4(6) + 6
= 30
B). G-CO.1 Benchmark
Assessment
Given the figure on the right,
Classify each angle as acute,
right, or obtuse
A. XTS
B. WTU
C. K is in the interior of
LMN, mLMK
=52°, and mKMN =
12°. Find mLMN.
D. BD bisects ABC,
mABD =
and mDBC = (y +
4)°. Find mABC.
E. Use a protractor to
draw an angle with a
measure of 165°.
G-CO.1 (C).
Understanding perimeter and area of
polygons and area circumference of
circles: Students apply formulas for
perimeter, area, and circumference.
In a circle a diameter is a segment that
passes through the center of the circle
and whose endpoints are on a circle. A
radius of a circle is a segment whose
endpoints are the center of the circle and
a point on the circle. The
circumference of a circle is the distance
around the circle.
The ratio of a circle’s circumference to
its diameter is the same for all circles.
This ratio is represented by the Greek
letter  (pi). The value of  is irrational.
Pi is often approximated as 3.14 or
.
Exercise problem on the right.
Find the circumference and area of a circle
with radius 8 cm. Use the  key on your
calculator.
Round the answer to the nearest tenth.
 50.3 cm
 201.1 cm2
G-CO.1 (C). Benchmark assessment
on the right: Understanding perimeter
and area of polygons and area
circumference of circles:
Find the area and perimeter of each:
1)
2).
Find the circumference and area of
each circle. Leave answers in terms
of .
3. Radius 2 cm
4. Diameter 12 ft
5. The area of a rectangle is 74.82 in2,
and the length is 12.9 in. Find the width.
G-CO.1 D
Develop and apply the formula for
midpoint. Use the Distance Formula and
the Pythagorean Theorem to find the
distance between two points in the
coordinate plane.
Example of finding the midpoint using
the midpoint formula
= (-5,5)
Find FG and JK.
Then determine whether FG  JK.
Exercise 2 on the right to find
distances of FG & JK
Step 1 Find the coordinates of each
point.
F(1, 2), G(5, 5), J(–4, 0), K(–1, –3)
Step 2 Use the distance formula
Pythagorean Theorem explanation
Pythagorean Theorem example
A = 5, b = 9 and substitute these into the
formula c2 = a2 + b2
= 52 + 92
= 25 +81
c2= 106
C = 10.3
G-CO.1D Benchmark assessment to
the right
1. The coordinates of the vertices of
∆ABC are A(2, 5), B(6, –1), and
C(–4, –2). Find the perimeter of
∆ABC, to the nearest tenth.
2. K is the midpoint of HL. H has
coordinates (1, –7), and K has
coordinates (9, 3). Find the
coordinates of L.
3. Find the lengths of AB and CD
and determine whether they are
congruent.
G-CO.3
In this cluster, when given a
rectangle, parallelogram,
trapezoid, or regular polygons,
students will describe the
rotations and reflections that
carry it onto itself.
G-CO.4
Students will develop definitions
of rotations, reflections, and
translations in terms of angles,
circles, perpendicular lines,
parallel lines, and line segments.
Rotation of 900 on the
coordinate plane.
Reflection across the x
axis.
Translation
G-CO.3 & G-CO.4
Benchmark assessment (right)
G-CO.5 and G-CO.2
When students are given a geometric
figure and rotation, reflection, or
translation, they will draw the
transformed figure (marked with primes)
using graph paper or geometry software.
In addition to the prime marks to
indicate a transformation, students will
also use the arrow notation.
Furthermore, they will stretch figures
using dilation techniques.
1. What transformation do the
wings of a butterfly suggest?
2. Find a synonym for each of the
following terms:
a. Rotate
b. Reflect
c. Translate
Example of translation with proper
notation
Example of a 90° rotation, ∆ABC 
∆A’B’C’
A transformation is a change in the
position, size, or shape of a figure. The
original figure is called the preimage.
The resulting figure is called the image.
A transformation maps the preimage to
the image. Arrow notation () is used to
describe a transformation, and primes (’)
are used to label the image.
Example of a reflection across the x axis
with proper notation:
DEFG  D’E’F’G’
Students are given vertices of a preimage
and an image and are asked to graph
them on the coordinate plane and
determine the transformation.
Exercise 1
A figure has vertices at A(1, –1), B(2,
3), and C(4, –2). After a transformation,
the image of the figure has vertices at
A'(–1, –1), B'(–2, 3), and C'(–4, –2).
Draw the preimage and image. Then
identify the transformation.
Answer
The transformation is a reflection across the
y-axis because each point and its image are
the same distance from the y-axis.
Exercise 2
A figure has vertices at E(2, 0), F(2, -1),
G(5, -1), and H(5, 0). After a
transformation, the image of the figure has
vertices at E’(0, 2), F’(1, 2), G’(1, 5), and
H’(0, 5). Draw the preimage and image.
Then identify the transformation.
Answer
The transformation is a 90°
counterclockwise rotation.
To find coordinates for the image of a
figure in a translation, add a to the
x-coordinates of the preimage and add b
to the y-coordinates of the preimage.
Translations can also be described by a
rule such as (x, y)  (x + a, y + b).
Exercise 3
Find the coordinates for the image of
∆ABC after the translation (x, y)  (x +
2, y - 1). Draw the image.
Step 1 Find the coordinates of
∆ABC.
The vertices of ∆ABC are A(–4, 2), B(–3,
4), C(–1, 1).
Step 2 Apply the rule to find the vertices
of the image.
A’(–4 + 2, 2 – 1) = A’(–2, 1)
B’(–3 + 2, 4 – 1) = B’(–1, 3)
C’(–1 + 2, 1 – 1) = C’(1, 0)
Step 3 Plot the points. Then finish
drawing the image by using a
straightedge to connect the vertices.
Answer to the right.
This rule can be applied to other
polygons.
Writing rules for transformations is
working backwards from when given a
rule to find an image.
Exercise 4
Use the diagram below to write a rule
for the translation of square 1 to square
3.
For example using the diagram on the
right, we follow these steps:
Step 1 Choose two points.
Choose a Point A on the preimage and a
corresponding Point A’ on the image. A
has coordinate (3, 1) and A’ has
coordinates (–1, –3).
Step 2 Translate.
To translate A to A’, 4 units are
subtracted from the x-coordinate and 4
units are subtracted from the
ycoordinate. Therefore, the translation
rule is (x, y)  (x – 4, y – 4).
A
’
G-CO.5 Benchmark Assessment
On the right, we have the benchmark
assessment that is related to all prior
activities and is closely parallel to the
standards. (refer to lesson plan)
1. A figure has vertices at X(–1, 1),
Y(1, 4), and Z(2, 2). After a
transformation, the image of the
figure has vertices at X'(–3, 2), Y'(–
1, 5), and Z'(0, 3). Draw the
preimage and the image. Identify the
transformation.
a. Take the preimage and
transform it 900 and write
its new vertices.
2. What transformation is suggested by
the wings of an airplane?
3. Given points P(-2, -1) and Q(-1, 3),
draw PQ and its reflection across
the y-axis.
4. Find the coordinates of the image of
F(2, 7) after the translation (x, y) 
(x + 5, y – 6).