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David Lovi

Transformations Unit Plan

Table of Contents

I. Content and Objectives

a. Description…………………………………………………………………...........2

b. Standards and Learning Objectives……………………………………………….3

II. Pre-Assessment and Analysis

a. Pre-Quiz…………………………………………………………...………………5

b. Analysis……………………………………………………………….…………...6

III. Lesson Plans

a. Translations………………………………………………………………………..8

b. Translations (using vectors)……………………………………………………...17

c. Reflections……………………………...………………………………………..28

d. Reflections (over all horizontal and vertical lines)……………………..………..37

e. Rotations………………………………………………………………...……….44

IV. Formative Assessments…………………………………..…………………………..51

V. Post-Assessment

a. Analysis…………………………………………………………………………..70

b. Post-Quiz…………………………………………………………………..……..72

c. Student Learning Examples………………………………………………...……73

d. Graphs……………………………………………………………………...…….82

VI. Reflection……………………………………………………………………...……..83

1. Context and Objectives

a. This unit will be taught to 8th

grade students in Marlboro Middle School.

Although the Marlboro school district is going through an unfortunate budget crisis,

the classroom the unit will be conducted in is very modern. The room is equipped

with new desks, a Smartboard, and an ELMO which are both used for each lesson in

the unit plan.

There are three Math 8 sections (period 3,4, and 6) that the unit plan will be used

for. Period 3 consists of 9 female students and 7 males (1 Hispanic and one African

American). Period 4 consists of 13 female students and 10 males (5 African

American, 2 Hispanic, 1 Asian, and 1 Middle Eastern student). Period 6 consists of

13 female students and 9 males (2 African American and 1 Hispanic). Period 6 is

also an inclusion class which is co-taught with a special education teacher. Six of the

students in this classroom are classified. Two of these students are classified as

having a learning disability, one is classified as other health impaired (OHI), one is

classified with an emotional disturbance, one is classified with a speech disorder, and

the last student’s classification has not been disclosed to the special education

department yet. However, the last student does require an aide to sit next to him in

the classroom. All students in this class are either age 13 or 14.

Based on the makeup of each period, there are a number of changes I will make to

my instruction. First, there are significantly more females in each period. I will make

sure to include each gender evenly. I will prevent both the female and male students

from feeling excluded in the class.

For period 6, it is important that I work closely with the co-teacher in order to

ensure that each special education students are receiving an equal opportunity for

their education and that they are treated equally.

b. The learning objectives for this unit are based primarily from the second module of

New York State’s adaptation of the Common Core Standards.

The Concept of Congruence

Topic A: Definitions and Properties of the Basic Rigid Motions

Lesson 2: Definition of Translation and Three Basic Properties

Lesson 3: Translating Lines

Lesson 4: Definition of Reflection and Basic Properties

Lesson 5: Definition of Rotation and Basic Properties

Lesson 6: Rotations of 180 Degrees

Topic B: Sequencing the Basic Rigid Motions

Lesson 7: Sequencing Translations

Lesson 8: Sequencing Reflections and Translations

Lesson 9: Sequencing Rotations

These 8 lessons from the Common Core module were synthesized into my 5 lesson unit plan by

members of the Math Department in Marlboro Middle School.

The lessons in this unit address the following focus standards and foundational standards.

Focus Standards

Understand congruence and similarity using physical models, transparencies, or geometry

software.

8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations:

a. Lines are taken to lines, and line segments to line segments of the same length.

b. Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be

obtained from the first by a sequence of rotations, reflections, and translations; given two

congruent figures, describe a sequence that exhibits the congruence between them.

Foundational Standards

Geometric measurement: understand concepts of angle and measure angles.

4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a

common endpoint, and understand concepts of angle measurement:

a. An angle is measured with reference to a circle with its center at the common endpoint of the

rays, by considering the fraction of the circular arc between the points where the two rays

intersect the circle. An angle that turns through 1/360 of a circle is called a “onedegree angle,”

and can be used to measure angles.

b. An angle that turns through one-degree angles is said to have an angle measure of degrees.

Draw and identify lines and angles, and classify shapes by properties of their lines and

angles.

4.G.A.1 Draw points, lines, line segments, rays, angles, and perpendicular and parallel lines.

Identify these in two-dimensional figures.

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or

perpendicular lines, or the presence or absence of angles of a specified size. Recognize right

triangles as a category, and identify right triangles.

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure

such that the figure can be folded along the line into matching parts. Identify line-symmetric

figures and draw lines of symmetry.

Math 8- Name

Geometry Pre-Quiz Date

Answer the following questions to the best of your ability.

The purpose of this assignment is to see what you know about geometry. Do your best

1. Label each quadrant on the graph below 2. Is the figure below symmetrical?

H

Yes or No

3. If we rotate the letter H from question two 360°, what will the new figure look like ?

4. What is the difference between a line and a line segment ? Draw an example of each

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=Hwrce0OsLvCvLM&tbnid=gph7zqqt5vW0DM:&ved=0CAgQjRwwAA&url=http://domathtogether.com/prob/random-walk-in-the-plane/coordinate-plane-2/&ei=npZDUpOhHZje4AOF04DgAw&psig=AFQjCNEoVSdGrkZrcEMFzEMz0PzFLKg9mQ&ust=1380247582555075

Analysis

This pre-assessment was designed to evaluate student’s prior knowledge of basic

transformations. Although each period was not introduced to the specific idea of

“transformations,” the students still had rudimentary knowledge of basic terms like symmetry,

rotation, lines, line segments, and the coordinate plane.

Each of the four items on the pre-assessment aligned to the objectives of the unit. Item one

shows how well students understand the coordinate plane. Specifically, I tested how well

students were able to label each of the four quadrants. Item two of the pre-assessment evaluates

the student’s knowledge of symmetry which corresponds to the reflection portion of the unit.

Item three of the pre-assessment identifies the student’s knowledge of rotation, which is another

portion of the unit. Item four of the pre-assessment targets the student’s knowledge of lines and

line segments. The last question corresponds to the translation portion of the unit.

Grading of the pre-assessment was based on a zero to four scale. Each question was worth

one point, and I did not award partial credit, so that the information I gathered is strictly based on

a student’s complete understanding of a topic. Throughout the grading process, there were many

opportunities for me to award partial credit. But my goal for the pre-assessment was to avoid

inflating the scores and attend to precision.

Out of the four items on the pre-assessment, number one and number four were the most

challenging for students. The information I collected from these two items greatly changed how

I taught the unit. What I learned most is that students had a great misunderstanding about the

names of the quadrants. Throughout the course of the unit, I made sure to incorporate ways of

continually assessing and reassessing this knowledge. Similarly, students also had a difficult

time understanding the difference between a line and a line segment.

More than half of the students scored in the lower end of the scale. Thirty-three students

scored in the 1-2 range and twenty-eight students scored in the 3-4 range. As I mentioned

earlier, one of the causes for the low scores was that I did not award partial credit. But in

general, the biggest reason for the low scores was a lack of understanding about the coordinate

plane and other basic geometric concepts.

Rather than see the grades of the pre-assessment as negative, I used these results as a positive

tool to improve this unit. Knowing what students have problems with, prior to a lesson, allowed

me the opportunity to address the problem areas and teach the new material. This pre-

assessment has been one of the most rewarding activities I have ever done in my entire teaching

career. I will certainly use this technique in the future.

David Lovi - Lesson Plan 1 of 5

Transformations (Translations)

Objective – Students will be able to translate polygons in the coordinate plane.

Pre-requisite skills

Students must have knowledge of the coordinate plane and know how to plot and

identify points in the form of (x,y) in the plane. Similarly, students must have knowledge

of lines, line segments, angles, and polygons such as triangles and quadrilaterals.

New skills to be taught

The new skills that will be taught are how polygons are translated in the coordinate

plane.

Vocabulary

Geometric Transformation - “A rule that assigns each point of the plane a unique point (P).” A simpler definition is ‘changing the position of a point on a coordinate plane.’

Translation - “A basic rigid motion that moves a figure along a given vector.” A simpler

definition is, ‘moving a shape in one direction from one place to another.’

Assessment of standards

I will gauge student understanding of this topic first through the results of their pre-

assessment.

<pre-assessment attached below>

I will gauge student understanding throughout the lesson through questioning and

homework and later on through their post test results. Exit slips may also be used to

gauge student understand depending on the amount of time left at the end of class.

<Exit slip attached below>

Resources – I will be using the white board and markers in the front of the classroom,

the pull-down coordinate plane attached to the white board, dry erase markers, an

ELMO (the projector in the classroom connected to the Smartboard), notes that will be

distributed to students <attached>, exit slips, homework sheets <attached>, and a

Smartboard if it is working properly.

Instructional Model

Introduction activity to activate prior knowledge

I will begin the lesson by greeting the class, taking attendance, collecting pre-

assessments, and asking the class if they had any comments about the assessment.

Since the students did not have homework from the previous school day, I will begin the

lesson by directing the class to the objective of the lesson “We will be able to translate

polygons in the coordinate plane.”

Modeling/Demonstration

After the introduction, I will distribute the notes to the classroom and begin a

brief discussion about what they think the coordinate plane is and what they think a

transformation means.

After questioning students about what a transformation is, I will provide them

with the definition of “geometric transformation” using the ELMO.

Next, I will introduce the idea of a mathematical “translation”, which is the first

transformation they will be learning. I will emphasize that a translation is just one of

three other transformations that they will be learning.

I will also explain that when we translate a polygon, the translated shape is

labeled differently than the original shape. I will demonstrate this rule through a brief

example on the board by translating a triangle ABC into the triangle A’B’C’.

Finally, I will state that in order to make transformations in the coordinate plane,

students must pay very close attention to the exact coordinates of the shape they are

transforming. I will demonstrate this idea by completing the Example 1 of the notes for

the class.

Guided practice with feedback

After modeling and demonstration of the first problem, I will ask the students if

they have any questions on the material and then clarify any misconceptions.

Next, students and I will begin to complete the other 3 exercises listed in the

notes handout and continue to clarify any misconceptions.

Using the pull-down coordinate plane on the board, I will have a student

complete the fourth example in front of the class.

Closure

To close the lesson, I will state that transformations are tools that

mathematicians use to alter polygons and that these polygons can be altered in a

number of ways. Next I will say that a translation is a mathematician’s way of moving

polygons in one direction. Finally, I will explain that translations are important because

they tell us exactly how we are moving a shape. I will then hand out the homework and

if there is time, I will distribute exit slips.

Adaptations for students with disabilities

To adapt this lesson for the students in my class with disabilities, I will pay close

attention to the notes they are taking in order to verify student participation and

understanding.

Extension

For students that complete the assignment early, I will ask them to develop and

graph a quadrilateral ABCD. After creating the polygon, they will translate the shape 10

units down and 30 units to the left and state coordinates of the image. They must

identify this new shape as A’B’C’D’

The following is the pretest that was given to students on prior to the first lesson of the unit.

Math 8- Name

Geometry Pre-Quiz Date


The purpose of this assignment is to see what you know about geometry. Do your best

2. Label each quadrant on the graph below 2. Is the figure below symmetrical?

H

Yes or No

3. If we rotate the letter H from question two 360°, what will the new figure look like ?

4. What is the difference between a line and a line segment ? Draw an example of each

The following are the notes that will be distributed to students for the lesson


1. The vertices of AB are A(-3, 4) and B(5, -6).

Translate AB to the left 3 units and down 4 units,

and determine the coordinates of B'A' .

2. The vertices of ABC are A(2, 5), B(4, 2),

and C(0, 2). Translate ABC to the

right 4 units and up 2 units,

and determine the coordinates of C'B'A' .

Math 8 - Name

NOTES: Translations Date

A geometric transformation is

.

One type of transformation is a translation. A translation is

.

Examples:

y

x

Examples:

Parallelogram MATH has the vertices M(-2, 1), A(-1, 3), T(3, 3), and H(2, 1). Draw the figure,

and its image under a Translation down 3 units. Label the image M’A’T’H’.

y

x

Triangle CAT has the vertices C(-3,1), A(2, 4), and T(3, 1). Draw the figure and its image under

a Translation down 6 units and 2 units to the right. Label the image C’A’T’.

y

x

The following is the homework that will be given to students after the lesson.

Math 8 – Name

Translations HW #1 Date

Directions: Use your knowledge of translations, and geometry in general, to answer the

following questions.

4. Triangle MNP is drawn below. Graph the image of this triangle after each of the

following translations on the grid below (there should be a total of 4

triangles on the grid!). Write the coordinates of each image’s vertices on the

lines provided.

y

M N

P

x

a) Left 2 units, down 2 units

b) Right 2 units, down 1 unit

c) Left 2 units, up 3 units

5.

The following is an exit ticket that may or may not be given to students at the end of this lesson

Math 8 – Name

Translations Exit Ticket Date


following question.

1. Draw a simple figure and name it “Figure A.” Draw its image

under some transformation, (i.e., translate “Figure A” anywhere

else on the page and name it Figure A’ ).

Describe, intuitively, how you moved the figure (describe the

transformation). Use a complete sentence.


Transformations (Translations using vectors)

Objective – Students will be able to translate polygons in a plane using vectors.


Students must have knowledge of the coordinate plane and know to how plot and

identify points in the form of (x,y) in the plane. Similarly, students must have knowledge

of lines, line segments, angles, and polygons such as triangles and quadrilaterals.

Finally, students must have knowledge about translations from the previous lesson.


The new skills that will be taught are what a vector is and how polygons are

translated in a plane using vectors.

Vocabulary

Vector – a line segment that indicates direction

Assessment of standards

I will gauge student understanding of this topic primarily through their performance

on the homework assignment from the previous lesson

I will gauge student understanding throughout the lesson through questioning, future

homework assignments, and through their post test and unit quiz. Exit slips may be

used to gauge student understand depending the amount of time left at the end of

class. <Exit slip attached>

Resources – I will be using the white board and markers in the front of the classroom,

the pull-down coordinate plane attached to the white board, dry erase markers, an

ELMO (the projector in the classroom connected to the Smartboard), notes that will be

distributed to students <attached>, exit slips, homework sheets <attached>, and a

Smartboard.

Instructional Model


I will begin the lesson by greeting the class, taking attendance, checking homework, and

asking the class if they had any questions or comments about the assignment from the

previous day. I will begin the lesson by directing the class to the objective of the lesson

“We will be able to translate shapes in the coordinate plane using vectors.”

To introduce the concept of vectors, I will draw a line, a line segment, and a ray on the

board. I will state that a vector is a ‘line segment with a direction’ and that it is ‘a ray

with an endpoint.’ Finally I will state that a vector is another way for mathematicians to

give directions on how to perform a specific translation.


After the introduction of the word, I will demonstration the translation of a

triangle using verbal and written instructions stating “two left, three up.” After

explaining and demonstrating this demonstration, I will state that we can translate the

same triangle to the same location by using a vector that also travels in the same

direction.

Next, using the notes we will discuss how the translated image of a polygon is

congruent to the original polygon. Similarly, we will discuss how the translation of a

polygon also preserves the length of sides and also creates a series of lines that are

parallel.

I will also explain that when we translate a polygon by using a vector, we can

describe the same motion in words and also by a rule in arrow notation in the form:

( x , y ) --> ( x + a , y + b )

where a and b are real numbers describing the motion of the vector in a plane.

Finally, I will remind students to properly label all of the points they graph for

both the original polygon, make sure that each letter on the vertex of the image

corresponds to the original image, and that the letters of the image are denoted prime

notation.


After modeling and demonstration of the objective, I will ask the students if they

have any questions on the material and I will clarify any misconceptions.

Next, students and I will begin to complete the first 3 exercises listed in the notes

handout and I will continue to clarify any misconceptions.

Students with then work independently on the multi-step vector translation

problem in the notes. Students will perform the vector translation of two points, an

angle, and a rectangle. I will circulate throughout the room and monitor student work,

answer questions, and clarify misconceptions. Finally, we will discuss their work and

answer the final three questions in the notes corresponding to the previous graph.

Closure

To close the lesson, I will state that vectors are just another tool that

mathematicians can use to describe how a polygon is translated. I will remind the class

that we can also describe a translation using words as well as arrow notation. Finally, I

will explain that even though we have three different ways of explaining a translation,

the image will still look the same. I will then hand out homework and if there is time, I

will distribute exit slips.



attention to the notes they are taking in order to monitor productivity and verify

student understanding.

Extension


graph a quadrilateral ABCD. After creating the polygon, students must create a vector

and then translate their quadrilateral using the vector they created. They must identify

this image as A’B’C’D’ and state the coordinates of the quadrilateral.

The following is the homework from lesson 1 that will be reviewed prior to lesson 2.

Math 8 – Name





following translations on the grid below (there should be a total of 4 triangles on the

grid!). Write the coordinates of each image’s vertices on the lines provided.

y

M N

P

x



c) Left 2 units, up 3 units (OVER)

5.


Math 8 - Name

NOTES: Translations with Vectors Date

Translations occur along a given vector:

1. A vector is .

2. One of its two endpoints is known as a ;

while the other is known simply as the .

3. Note the starting and endpoints of the following vectors:

A translation of a plane along a given vector is a

.

The Three Properties of Translations

(T1) A translation maps a line to a line, a ray to a ray, a segment to a segment, and an

angle to an angle.

(T2) A translation preserves .

(T3) A translation preserves .

Examples:

Translate line along the vector . What do you notice about and its image, ?

If and were translated along vector , describe the movement you would use to complete

this translation. If lines and are parallel, what do you know about their translated

images?

Translate , point , point , and rectangle along vector Sketch the images and label

all points using prime notation.

Answer the following questions using the graph from question 3:

a) What is the measure of the translated image of . How do you know?

b) Connect to . What do you know about the line formed by and the line containing the

vector ?

c) Connect to . What do you know about the line formed by and the line containing the

vector ?

d) Given that figure is a rectangle, what do you know about lines and and their

translated images? Explain.

The following is the homework that was given to students after this lesson .

Math 8 – Name




1. A rectangle has its vertices at M(1, 1), N(6, 1), O(6, 5), and P(1, 5). The rectangle is

translated according to AB . What are the coordinates of M’, N’, O’, and P’? Graph

rectangles MNOP and M’N’O’P’.

y A B x

2. Use arrow notation to write a rule that describes the translation of M’N’O’P’ to MNOP

drawn above.

(OVER)

3.

4.

5.

6.


Transformations (Reflections over the x or y axis)

Objective – Students will be able to reflect polygons in the coordinate plane over the x and y axis.


Students must have knowledge of the coordinate plane and know how plot points in

the form of (x,y) in the plane. Similarly, students must have knowledge of the concept

of transformations, translations, lines, line segments, angles, and polygons such as

triangles and quadrilaterals.


The new skills that will be taught are how polygons are reflected in the coordinate

plane (specifically over the x or y axis).

Vocabulary

Reflection “A basic rigid motion that moves a figure across a line.” This is the definition given by the Common Core module. I will state that a reflection is a motion that creates a mirror image, or a “flipped” image of a figure across a line. Resources – I will be using the white board and markers in the front of the classroom,

the pull-down coordinate plane attached to the white board, pieces construction paper

cut into triangles, dry erase markers, an ELMO (the projector in the classroom

connected to the Smartboard), notes that will be distributed to students <attached>,

exit slips <attached>, homework sheets <attached>, and a Smartboard.


I will begin the lesson by greeting the class, taking attendance, checking

homework, and asking the class if they had any problems or comments about the

assignment from the previous day. I will begin the lesson by reviewing the homework

assignment about translations using vectors. Next I will direct the class to the objective

of the day “We will be able to reflect shapes in the coordinate plane over the x and y

axis.”


After the introduction, I will distribute the notes handout to the classroom and

begin a brief discussion about what they think a reflection is in geometry.

After questioning students about what a reflection is, I will provide them with the

definition of a geometric “reflection” using the ELMO.

Next, I will introduce what a reflection looks like in the coordinate plane and

demonstrate this concept on the board by using two pieces of construction paper that

are cut into congruent triangles. I will demonstrate the reflection of a triangle first over

the y-axis. Next I will use the triangles to demonstrate a reflection over the x-axis. I will

emphasize that a reflection is another one of three other transformations that they will

be learning.

I will also explain that just as we did for a translation, the image must be labeled

properly. Using the triangles, I will demonstrate exactly where the corresponding letters

of the image should be after reflecting the triangle in either axis.

Finally, I will reiterate that when making transformations in the coordinate plane,

students must pay very close attention to the exact coordinates of the shape they are

transforming. I will demonstrate this idea by completing the first two examples of the

notes for the class.


After modeling and demonstration of the first examples, I will ask the students if

they have any questions on the material and clarify any misconceptions.

Next, students and I will begin to complete the other 3 exercises listed in the

notes handout and we will continue to clarify any misconceptions.

Using the pull-down coordinate plane on the board, I will have a student

complete the last example in front of the class.

Closure

To close the lesson, I will restate that we have now learned two of the three

different translations. I will remind the class that translations are used to slide a

polygon and that reflections are used to “flip” a polygon. Finally, I will emphasize that

the image after a translation and an image of a reflection will always look different

because a reflection changes the order of each vertex and their corresponding label. I

will then hand out homework and if there is time, I will distribute exit slips.





Extension


graph a triangle ABC in the first quadrant. After creating the polygon, they will reflect

the shape across the y-axis. They must identify this new polygon as A’B’C’. Next they

will reflect A’B’C’ in the x-axis. Finally they must identify this new polygon as A”B”C”.


Math 8 – Name




2. A rectangle has its vertices at M(1, 1), N(6, 1), O(6, 5), and P(1, 5). The rectangle is

translated to the left 4 units and down 3 units. What are the coordinates of M’, N’, O’, and

P’? Graph rectangles MNOP and M’N’O’P’.

y x


drawn above.


Math 8 – Name

NOTES: Reflections Date

A reflection is

Examples:

The coordinates of XY are X(-4, -4) The vertices of LMN are L(7, -5),

Y(8, 9). Reflect XYacross the x-axis M(4, -3), N(6, -1). Reflect LMN

and determine the coordinates of Y'X' . across the y-axis and determine the

coordinates of N'M'L' .

y y

x x

Practice:

3.

4.

5.

6.

The following is the homework that will be given at the end of the lesson

Math 8 – Name

Reflections HW #1 Date

Directions: Perform the line reflections for each of the following problems. Pay attention to

which axis you are to reflect over. Also, write the coordinates for both the original figures,

and the images.

1. Reflect triangle ABC over the y-axis. Draw the new triangle and label it as A’ B’ C’.

A __________ A’ __________

B __________ B’ __________

C __________ C’ __________

2. Reflect triangle DEF over the x-axis. Draw the image and label it as D’ E’ F’.

D __________ D’ __________

E __________ E’ __________

F __________ F’__________

The following is an exit ticket that may or may not be given to students at the end of this lesson

Math 8 – Name


Directions: Use your knowledge of reflections, and geometry in general, to answer the

following question.

1. Let there be a reflection across line segment . Reflect Δ and label the reflected image.

2. Use the diagram above to state the measure of on ( ). Explain. 3. Use the diagram above to state the length of segment on ( ). Explain.


Transformations (Reflections over horizontal and vertical lines)

Objective – Students will be able to reflect polygons over all horizontal and vertical lines.





triangles and quadrilaterals. Specifically, student must have knowledge of reflections

over the x and y axis.


The first set of new skills that will be taught are writing the equation of horizontal

and vertical lines and the ability to identify and sketch horizontal and vertical lines based

on their corresponding equation.

The last new skill that will be taught to students is to understand are how polygons

are reflected and labeled in the coordinate plane over all horizontal and vertical lines.

Vocabulary

Reflection - “A basic rigid motion that moves a figure across a line.” This is the definition given by the Common Core module. I will state that a reflection is a motion that creates a mirror image, or a “flipped” image of a figure across a line. Resources – I will be using the white board and markers in the front of the classroom,

the pull-down coordinate plane attached to the white board, pieces construction paper

cut into triangles, dry erase markers, an ELMO (the projector in the classroom

connected to the Smartboard), notes that will be distributed to students <attached>, a

problem set, homework sheets, and a Smartboard.


I will begin the lesson by greeting the class, taking attendance, collecting work,

and asking the class if they had any problems or comments about the assignment from

the previous school week. I will begin by asking the class what a reflection over the x

and y axis looks like. I will have a student demonstrate using the construction paper

triangles from the previous lesson. Next I will direct the class to the objective of the day

“We will be able to reflect polygons over all horizontal and vertical line.”


After the introduction, I will distribute the notes handout to the classroom and

begin a brief discussion about what how we identify the equation of both a horizontal

line and a vertical line in the coordinate plane. I will state that the equation of a

horizontal line has the form:

Y = a

Where a is the point at which the horizontal line crosses the y axis.

Similarly, I will state that the equation of a vertical line has the form

X = a

Where a is the point at which the vertical line passes through the x axis.

After demonstrating a few examples of what various horizontal and vertical lines

are, I will ask students what the equation of each axis will be students will be, and

students record this information in the notes handout.

Next, I will introduce what a reflection of a triangle looks over the vertical line

x=3 looks like compared to a reflection of the same triangle. Similarly, I will repeat this

process for the same triangle but for a reflection over the line y=3 compared to a

reflection over the x axis. I will reiterate that a reflection is another one of three other

transformations that they will be learning.

I will also explain that just as we did for a translation, the image must be labeled

properly. Using the triangles, I will demonstrate exactly where the corresponding letters

of the image should be after reflecting the triangle in either axis.


students must pay very close attention to the exact coordinates of the polygons they are

transforming. I will demonstrate this idea by completing the first two examples of the

notes for the class.

Guided practice and independent practice with feedback

After modeling and demonstration, I will ask the students if they have any

questions on the material and clarify any misconceptions.

Next, students and I will begin to complete the other exercises listed in the

problem set of the second notes handout and I will continue to clarify any

misconceptions.

I will alternate between completing problems with the class using the ELMO, the

pull-down coordinate plane on the board, and allowing the students to complete the

problems independently while circulating throughout the room to monitor

understanding.

Closure

To close the lesson, I will restate that we have now learned two of the three

different translations. I will remind the class that translations are used to slide a

polygon and that reflections are used to “flip” a polygon. Next I will emphasize that the

image after a translation and an image of a reflection will always look different because

a reflection changes the order of each vertex and their corresponding label. Finally I will

state that students must continue to reinforce their knowledge of coordinate points,

horizontal lines, and vertical lines by studying to make sure they all know the proper

points they are reflecting and the proper line they are reflecting over. I will then hand

out homework.





Extension


graph a triangle ABC in the first quadrant. After creating the polygon, they will reflect

the shape across a vertical line in the coordinate plane other than the y-axis. They must

identify this new polygon as A’B’C’. Next they will reflect A’B’C’ over a horizontal line in

the coordinate plane other than the x-axis. Finally they must label this new polygon as

A”B”C”.


Math 8 – Name




and the images.


A __________ A’ __________

B __________ B’ __________

C __________ C’ __________


D __________ D’ __________

E __________ E’ __________

F __________ F’ __________


Math 8- Name

NOTES: Horizontal and Vertical Lines Date

Yesterday, we performed over the x- and y-axis. There

are other horizontal and vertical lines on the coordinate plane, however.

Vertical Lines

The equations of all vertical lines are in the form x = a, where a is the point that the line

crosses the .

Horizontal Lines

The equations of all vertical lines are in the form y = a, where a is the point that

the line crosses the .

Practice: Identify the equations of each of the following lines drawn on the coordinate plane.

1. y 2. y 3. y x x x


Transformations (Rotations)

Objective – Students will be able to rotate polygons in the coordinate plane.





triangles and quadrilaterals. Specifically, student must have knowledge of the process

of measuring the angle of rotation by using degrees.


The first set of new skills that will be taught are writing the equation of horizontal

and vertical lines and the ability to identify and sketch horizontal and vertical lines based

on their corresponding equation.

The last new skill that will be taught to students is to understand are how polygons

are reflected and labeled in the coordinate plane over all horizontal and vertical lines.

Vocabulary

Rotation - a transformation that turns a figure about a point called a center of rotation. Resources

I will be using the white board and markers in the front of the classroom, the pull-down

coordinate plane attached to the white board, pieces construction paper cut into

triangles, dry erase markers, an ELMO (the projector in the classroom connected to the

Smartboard), notes that will be distributed to students, homework sheets <attached>,

and a Smartboard.


I will begin the lesson by greeting the class, taking attendance, collecting work,

and asking the class if they had any problems or comments about the assignment from

the previous school week. I will begin by asking the class to state the previous

transformations we have discussed (translations and reflections). After writing these on

the board, I will direct a student to read the learning target of the day. “We will learn

how to rotate polygons in the coordinate plane.” Next I will state that rotations are the

third transformation we will learn about that creates images that are congruent to the

original image. Next, I will demonstrate the idea of a rotation about the origin by using

the construction paper triangles from the previous lesson. Next I will direct the class to

stand up and face me. Next we will practice making rotations of varying degree (90,

180, 270, and 360) by rotating our bodies.


After the introduction, I will distribute graph paper to the classroom and begin a

brief discussion about how we further describe and depict rotations of polygons in the

coordinate plane. I will create a triangle RST with coordinates:

R (0,0) , S(4,3) , T(4,0)

After plotting the points, I will ask the class to state what quadrant the image

R’S’T’ will be in after a rotation of 90 degrees. The image and method will be

demonstrated to the class by rotating the paper 90 degrees and examining exactly

where the rotation will put the transformation. Similarly, the class will rotate RST and

create R’S’T’.

Similarly, I will ask the class and I will repeat this same procedure for a rotation of

180 and 270 degrees creating the image of R’’S’’T’’ and R’’’S’’’T’’’.

Finally, the class will state what the image of a 360 degree rotation will look like

Next, I will state that a positive rotation is a counter clockwise motion and that

negative rotations occur in a clockwise motion. I will indicate that a positive rotation

occurs in the same order as the quadrants are numbered.

I will also explain that just as we did for translations and reflections, the images

of a rotation (or rotations) must be labeled properly. Using the construction paper

triangles, I will demonstrate exactly where the corresponding letters of the image

should be after reflecting the triangle in either axis.


students must pay very close attention to the exact coordinates of the polygons they are

transforming. I will also state the rules for finding the coordinates of an image for

rotations of 90, 180, or 270. I will stress that students now may find the location of an

image by using a visual method or by finding the coordinates of the image first.

Guided practice and independent practice with feedback

After modeling and demonstration, I will ask the students if they have any

questions on the material and clarify any misconceptions.

Next, students will plot the points of the quadrilateral MNOP with coordinates

M(1,6) , N(5,6) , O(4,2) , P(2,2). After students map MNOP, they will rotate the polygon

90 degrees and 180 degrees and create the image of M’N’O’P’ and M’’N’’O’’P’’. While

students work on this problem, I will circulate throughout the classroom and continue to

clarify any misconceptions.

Closure

To close the lesson, I will restate that we have now learned three of the three

different translations that create congruent images. I will remind the class that

translations are used to “slide” a polygon; reflections are used to “flip” a polygon; and

rotations are used to “spin” polygons. Finally I will state that students must continue to

reinforce their knowledge of coordinate points and degrees of measurement by

studying to make sure they all know the proper points they are rotating, know how

much they are rotating, and know the proper direction they are rotating. I will then

hand out homework.





Extension

For students that complete the assignment early, I will ask them to state what

the rule for a rotation of 360 degrees is in arrow notation. Next I will make them find 3

different degrees of rotation that equal 90, 180, and 270. For example, 180 = 540 =900

= 1260.

The following are the homework assignments that will be distributed to students

for this lesson.

Formative Assessment

Table of Contents

Translations HW #1…………………………………………...…52

Translation Exit Ticket #1……………………………………….54

Translations HW #2…………………………………………..….55

Translations “Do Now”…………………………..………………57

Translation Exit Ticket #2……………………………………….58

Reflections HW #1 ………………………………………………59

Rotations HW #1……………………………………………...….60

Rotations HW #2……………………………………………...….62

POW (Problem of the Week) #1………………………………....64

POW #2………………………………………………………..…66

Analysis………………………………………………………………..68

Math 8 – Name





following translations on the grid below (there should be a total of 4 triangles on the

grid!). Write the coordinates of each image’s vertices on the lines provided.

M N

P



c) Left 2 units, up 3 units (OVER)

Math 8 – Name


Directions: Use your knowledge of translations, and geometry in general, to answer the following

question.

2. Draw a simple figure and name it “Figure A.” Draw its image under

some transformation, (i.e., translate “Figure A” anywhere else on the

page and name it Figure A’ ).

Describe, intuitively, how you moved the figure (describe the

transformation). Use a complete sentence.

Math 8 – Name




3. A rectangle has its vertices at M(1, 1), N(6, 1), O(6, 5), and P(1, 5). The rectangle is translated according to AB . What are the coordinates of M’, N’, O’, and P’? Graph rectangles MNOP and M’N’O’P’.

y A B x


drawn above.

(OVER)

3.

4.

5.

6.

10/3/13

DO NOW

Write the other two ways that we can translate a shape.

1. Written instruction (for example Left 2, Down 3)

2.

3.

Write your answers on a separate sheet of paper.

Please include your name and date on the Do Now.

Math 8 – Name


Directions: Use your knowledge of reflections, and geometry in general, to answer the following

question.

1. Let there be a reflection across line segment . Reflect Δ and label the reflected image.

2. Use the diagram above to state the measure of on ( ). Explain. 3. Use the diagram above to state the length of segment on ( ). Explain.

Math 8 – Name




and the images.


A __________ A’ __________

B __________ B’ __________

C __________ C’ __________


D __________ D’ __________

E __________ E’ __________

F __________ F’ __________

Math 8 - Name

POW #1: Geometric Transformations I Date

Directions: Each question on this POW is worth 10 points, making this assignment worth a total

of 20 points. Draw all figures with a RULER AND PENCIL!

1.

2. On the coordinate plane below, draw the image of polygon ABCDE according to JK .

Label the image A’B’C’D’E’.

K

J

Math 8 – Name

POW #2: Geometric Transformations II Date

Each question on this POW is worth 10 points, making this assignment worth a total of 20 points.

1. Melissa drew figure HIJK on the coordinate plane below. Reflect Melissa’s figure over

the line x = -2 and label the image appropriately.

The formative assessments of this unit consist of: five homework assignments, two exit

tickets, a “Do Now,” and two “Problems of the Week (POW).” Each of these assessments

played an important role in the development of both lesson planning as well as instruction.

A homework assignment was given at the end of each lesson in the unit. Each student

completed the assignment to the best of their ability and presented their work at the beginning of

each lesson. I would circulate throughout the room and verify that each student completed, or

attempted to complete, the assignment. Through the process of reviewing every students work, I

was able to tell which problems the students had difficulties with. I would then review the

homework with the class and emphasize the problem areas that many had and clarify any

misconceptions. Similarly, the problem areas of the homework would continue to be addressed

throughout the lesson.

Exit slips were developed for two of the five lessons in this unit. Ideally, I would complete

the instructional portion of the lesson with enough time to distribute the slips and have the

students complete them. Unfortunately, for these two lessons, I did not have enough time to

distribute the slips so instead I projected the exit slips onto the Smartboard and had students

respond to the questions verbally. The responses from the exit slips were a way of providing me

immediate feedback about student’s understanding of the lesson from that day. After receiving

this feedback, I was able to plan a way to address their misconceptions in the following lesson.

Also, the feedback allowed me to alter the lesson of that day for the following periods of Math 8.

The Do Now assignment was used in the beginning of lesson 3 and was a brief informal

assessment about students understanding of translations from the first two lessons. The

information I received from the Do Now gave me the opportunity to see how well students have

comprehended the information of the unit without having to give a formal quiz. After reflecting

on the responses from the Do Now assignment, I altered instruction by highlighting the key

concepts in a different way than I had been. Aside from being a reminder to students that they

need to retain the specific information from the beginning of the unit, their Do Now responses

showed me the strengths of my students and allowed me to identify the things I have been

teaching well.

Out of all of the formative assessments that I used in this unit, the Problems of the Week were

the two assignments that were graded. Although all students’ participation for the homework

assignments was recorded in the grade book, there were no quantitative results recorded. Each

of the POW assignments was graded and worth a total of twenty points each. These particular

assessments were valuable tools in the unit because they reflected both student understanding

and work ethic. Each POW was distributed and students were given one week to complete the

assignment. Many students started the assignment early and made sure that they would receive

full credit while others carelessly completed the assignment. After grading each of the

assignments, I was able to see the areas of the unit that students still have trouble with and need

further instruction on. After grading these assignments, I also used the information to

incorporate time in the lesson to stress a proper work ethic for all assignments as well as clearly

state what I am expecting for future assessments.

After comparing both pre and post-assessment data, it is clear that students gained a deeper

understanding of the coordinate plane, reflections, rotations, and translations. Each period of

Math 8 made improvements from their initial pre-assessment. Although there were a few

students who received lower grades on the post assessment, the majority of the class made

improvements in the four objectives that were being assessed. These areas were general

knowledge of the coordinate plane, reflections, rotations, and translations (specifically using

vectors). Each of these four items corresponded to the four questions given on the pre-

assessment.

The biggest improvement by students on the post quiz was that almost every student was able

to identify the quadrants in the coordinate plan correctly. Properly labeling quadrants was one

the biggest area that students struggled with on the pre quiz. During my instruction of the unit, I

made sure to emphasize what quadrant we were working in.

The next largest improvement by students on the post quiz was that students had a much

better understanding of what a line, a line segment, and a vector were. Since translations using

vectors was a major topic in the unit, I needed to assess their understanding of the concept.

However, after grading the assessments, it is clear that the students still had a number of

misconceptions about what a vector is.

One of the biggest challenges students had on the post quiz that was not an issue for the pre-

assessment was rotations. Students performed very poorly on item 3 despite having covered

rotations for multiple days. Oddly enough, the rotation question on the pre-assessment did not

cause as many problems as the item from the pre-quiz.

For the reflection question, there were an average number of correct answers compared to the

pre-quiz. However the question was not specifically worded as a reflection question and may

have caused some confusion.

Out of the 64 students that took the pre-assessment, 54 students took the post-assessment. Of

these 54 students, 53 of them took both the pre and post quiz. Out of the 53, 11% had a decrease

in score, 40% remained the same, and 49% increased their score. However, since the post quiz

had a higher difficulty level than the pre-quiz, I consider the student’s scores remaining the same

to be equivalent to an increase in score.

In the future, I will spend more time focusing on the rotation aspect of this unit. Based on

both the pre and post assessments, students have a hard time understanding the concept of

rotation and measuring angles in degrees. Similarly, I will focus more on the definition of a

vector.

Math 8- Name

Geometry Post-Quiz Date


This assignment is to see what you learned about geometry this week. Do your best

3. Draw a triangle in the first quadrant. 2. Is the figure below symmetrical?

Draw a rectangle in the fourth quadrant.

Draw a circle in the third quadrant. B

Yes or No

If yes, draw exactly where the figure is symmetrical.

3. Spin the letter B from question two 180°. Sketch the new figure .

Is this process a transformation ? If so, what kind of transformation?

4. Draw: a line, a line segment, and a vector

Explain why a vector is different from a line segment.


The following student struggled with the pre quiz and continued to struggle with the post quiz.

The following student displayed an average performance on the pre quiz and the post quiz.

The following student showed a great deal of improvement from the pre quiz to the post quiz.

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Reflecting on my instruction of this unit plan based on the assessments, there are a number of

things that were successful and there are a number of things that I would change. The

information that I gained through the formative assessments of the unit, as well as informal

assessments from each lesson, gave me very concrete data to improve not only my instruction of

this unit in the future, but also my instruction of any math class in the future. Through the

process of developing and implementing this unit plan, I experienced firsthand the effects and

importance of prior knowledge. Reflecting on the assessment data I collected from a unit on

geometric transformations, I know that I will be able to apply the information to every unit that I

teach in the future.

All of the assessments in the unit provided me with very unique and specific information that

I have used during the unit and will continue to use in the future as a math teacher. One of the

biggest lessons that I learned through the process of this lesson plan is that I need to assess prior

knowledge immediately. Although the pre-assessment was an invaluable tool for indicating

prior knowledge, the information about student’s general knowledge about geometry would have

been much more useful to me if I had gathered this data much earlier in the year. What I learned

through instruction of each lesson in the unit is that students need to be taught things multiple

times. For example, the first item on my pre-assessment assessed the student’s knowledge about

the quadrants of the coordinate plane. Although this is knowledge that they should have gained

from 6th

and 7th

grade, many students still had trouble identifying where the first quadrant is

located and exactly what a line or line segment is. If I had known about this problem earlier, I

would attempted to clarify these misconceptions earlier and adjusted the unit plan prior to the

implementation rather than having to change it during its implementation. Unfortunately, even

after almost two weeks of reinforcing these ideas, there were still some students that had trouble

with these ideas based on the post assessment.

One of the biggest changes that I would make to this unit on transformations is introduce the

students to all of the three transformations sooner. The pre-assessment did assess all three areas

of the unit. But there were certain concepts, specifically rotations, which I would have like to

have discussed much earlier than they were in the unit. In the plan, rotations were the last topic

to be covered. Although the rotation question on the pre-assessment did not appear to be a

problem for most students, teaching the actual lesson showed me that many students had a very

poor understanding of how to rotate an object when given an angle measure in degrees. This

aspect of rotations is another geometric concept that should be prior knowledge and if I were to

teach this unit again, I would put more emphasis on this concept.

Completing this unit plan has been one of the most rewarding assignments I have ever done in

college. Every step of the plan was a learning experience and I gained valuable data on how to

improve my teaching. Developing, teaching, assessing, and reflecting on the unit were all unique

components of the plan, and each step had varying levels of success that are continuing to help

me become a better teacher.

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