Curriculum Development Project

Unit: Parallel and Perpendicular Lines (Week 5 – Week 7)

A. Number of Lesson: 4 Lessons

1. Lines and Angles

2. Angels Formed by Parallel Lines and Transversal

3. Proving Lines Parallel

4. Perpendicular Lines

B. Grade Level: 10th Grade

C. Rationale: The study of geometry teaches not only geometry but the systematic

dissection and construction of a problem.  This unit codifies knowledge from previous

courses of study into a firm base that will be used to further study geometry.

D. Standards:

1. G.CO.9. Prove theorems about lines and angles.

2. G.CO.12. 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 point not on the line.

E. Goals:

1. Identify parallel, perpendicular, and skew lines

2. Identify the angles formed by two lines and a transversal classify pairs of angles

3. Find angle measures by solving systems of equations

4. Prove and use theorems about the angles formed by parallel lines and a transversal

5. Prove two lines are parallel using the angles formed by a transversal

6. Determine whether lines are parallel, perpendicular or neither

7. Prove and apply theorems about perpendicular lines

F. Big Ideas/Essential Understanding: In fields such as architecture and construction, you

must ensure that opposite walls of a building are parallel and adjacent walls are

perpendicular (otherwise your house would fall down).

G. Essential Questions:

1. How do alternate exterior angles differ from alternate interior angles?

2. In a two-column proof, what can be written under the reasons column?

3. What can you conclude about two coplanar lines that are perpendicular to the same line?

4. If you know two points on line p and two points on line q, how could you tell that p is

parallel to q?

5. What are the methods to prove two lines are parallel?

H. Resources:

1. Teacher's Power Point

2. Smart Board and White Board

3. Markers

4. Geometry Concepts and Skills Textbook

5. On-Line Version of Textbook

6. Handouts

I. Lesson:

1. Lesson 1: Lines and Angles

Objectives: Identify parallel, perpendicular, and skew lines. Identify the angles formed

by two lines and a transversal.

Standards: G.CO.1. Know precise definitions of angle, circle, perpendicular line,

parallel line, and line segment, based on the undefined notions of point, line, distance along a

line, and distance around a circular arc.

Materials needed:

-projector screen with Geo-Gebra, if these are not available the teacher can do the constructions

manually on the board.

-graph paper,

-compass (1 per student)

-ruler (1 per student)

-pictures of Euclid and Descartes (attached), to add interest to the history portions of the lesson

Lesson Procedure

Motivation: History of Euclidean Geometry. **Photo of Euclid attached at end of lesson.

Give a brief history of where most of our modern-day geometry comes from:

Euclid of Alexandria, a Greek mathematician from around 300 BC is called the “Father of

Geometry”. Most of the geometry we are studying today, called Euclidean Geometry, comes

from Euclid’s book, called Elements. This means that we are studying mathematics that was

discovered over 23 centuries ago! That’s 2300 years.

We know very little about Euclid’s life. Most of what we know about him was written centuries

after he lived. His birth and death dates and locations are unknown, we just estimate them. We

also have no idea of the circumstances surrounding his life. We do not even know what he

looked like but yet he is one of the reasons we are studying parallel lines today! The photo of

Euclid is completely of the artist’s imagination!

Warm Up: attached in the PPT. Please click here.

Discussion: What does it mean for two lines to be parallel?

a. Lines that never intersect.

b. Lines that stay the same distance from one another for infinity.

What does it mean for two lines to be perpendicular?

c. Lines that intersect at right angles.

d. ***This would be an appropriate time to review what a right angle is in order to

refresh students’ memories.

Also, key terms will be discussed and examples will be solved with students. (Definitions and

examples are in the PPT.) Please click here.

Activity: (This can be done in pairs, groups, or solo.)

a. Have students draw two parallel lines cut by a transversal on their paper.

b. Have them label all of the angles created (1-8, A-H, etc.)

c. Have the students write down everything they know about the angles. They can

use arcs to label congruent angles and also identify angles that are supplementary,

adjacent, etc.

d. Students can compare their answers with a partner. This is a great time to talk

about how when you label things differently than someone else you come up with

the same results but it is often difficult to compare!

Differentiated instruction: Students will be dived into three groups and depends on their

level they have different practice questions. (Worksheets are attached) Please click here.

Wrap up: Students answer the following question in complete sentences on a sheet of

paper. Explain your learning out of this lesson in a paragraph. (Exit Ticket) Please click here.

Assessment strategies: Lesson quiz will be given to students at the end of the lesson.

(Lesson Quiz is in the PPT.) Please click here.

Photos for history in Parallel Lines Lesson

Euclid

Descartes

2. Lesson 2: Angles Formed by Parallel Lines and Transversal

Objectives: Prove and use theorems about the angles formed by parallel lines and a

transversal.

Standards: G.CO.9. Prove theorems about lines and angles. Theorems include: vertical

angles are congruent; when a transversal crosses parallel lines, alternate interior angles are

congruent and corresponding angles are congruent; points on a perpendicular bisector of a line

segment are exactly those equidistant from the segment’s endpoints.

Materials needed:

-projector screen with Geo-Gebra, if these are not available the teacher can do the constructions

manually on the board.

-graph paper,

-compass (1 per student)

-ruler (1 per student)

-pictures of Euclid and Descartes (attached), to add interest to the history portions of the lesson

Lesson Procedure

Motivation: Who uses this? Piano makers use parallel strings for the higher notes. The

longer strings used to produce the lower notes can be viewed as transversal. (See example 3 in

the PPT.)

Warm Up: Attached in the PPT. Please click here.

Discussion: Tell the students that today we will be looking at parallel lines when they

are cut by a transversal. Also, corresponding angle postulate will be discussed and examples will

be solved with students. (Definitions and examples are in the PPT.) Please click here.

Differentiated instruction: Students will be dived into three groups and depends on their

level they have different practice questions. (Worksheets are attached) Please click here.

Wrap up: Students answer the following question in complete sentences on a sheet of

paper

Explain your learning out of this lesson in a paragraph. (Exit Ticket) Please click here.

Assessment strategies: Lesson quiz will be given to students at the end of the lesson.

(Lesson Quiz is in the PPT.) Please click here.

3. Lesson 3: Proving Lines Parallel

Objectives: Use the angles formed by a transversal to prove two lines are parallel.

Standards: G.CO.9. Prove theorems about lines and angles. Theorems include: vertical

angles are congruent; when a transversal crosses parallel lines, alternate interior angles are

congruent and corresponding angles are congruent; points on a perpendicular bisector of a line

segment are exactly those equidistant from the segment’s endpoints.

Materials needed:

-projector screen with Geometer’s Sketchpad,

-computer,

-graph paper,

-compass (1 per student)

-ruler (1 per student)

-pictures of Euclid and Descartes (attached), to add interest to the history portions of the lesson

Lesson Procedure

Motivation: Who uses this? Rowers have to keep the oars on each side parallel in order

to travel in a straight line. (See example 4 in the PPT.) Please click here.

Warm Up: Attached in the PPT. Please click here.

Activity: Investigation 1.2 in the following website will be done with students in the

computer lab. It is a Geometer’s Sketchpad activity. In this activity students will come up with

conjectures. (http://www.cc.k12.nf.ca/sketch/discover11.htm)

Discussion: Parallel line theorems will be discussed and examples will be solved with

students. (Theorems and examples are in the PPT.) Please click here.

Differentiated instruction: Students will be dived into three groups and depends on their

level they have different practice questions. (Worksheets are attached) Please click here.

Wrap up: Students answer the following question in complete sentences on a sheet of

paper

Explain your learning out of this lesson in a paragraph. (Exit Ticket) Please click here.

Assessment strategies: Lesson quiz will be given to students at the end of the lesson.

(Lesson Quiz is in the PPT.) Please click here.

4. Lesson 4: Perpendicular Lines

Objectives: Prove and apply theorems about perpendicular lines.

Standards: G.CO.9. Prove theorems about lines and angles. Theorems include: vertical

angles are congruent; when a transversal crosses parallel lines, alternate interior angles are

congruent and corresponding angles are congruent; points on a perpendicular bisector of a line

segment are exactly those equidistant from the segment’s endpoints.

Materials needed:

-projector screen with Geo-Gebra, if these are not available the teacher can do the constructions

manually on the board.

-graph paper,

-compass (1 per student)

-ruler (1 per student)

-pictures of Euclid and Descartes (attached), to add interest to the history portions of the lesson

Lesson Procedure

Motivation: Why learn this? Rib currents are strong currents that flow away from the

shore line and are perpendicular to it. A swimmer who gets caught on a rip current can get swept

far out to sea. (See example 3 in the PPT.) Please click here.

Warm Up: Attached in the PPT. Please click here.

Activity: Student will investigate the perpendicular lines before they learn the theorems.

http://www.geogebra.org/en/upload/files/UC_MAT/Teresa%20Munninghoff/

Parallel_Perpendicular_Lines.html

Discussion: Perpendicular line theorems will be discussed and examples will be solved

with students. (Theorems and examples are in the PPT.) Please click here.

Differentiated instruction: Students will be dived into three groups and depends on their

level they have different practice questions. (Worksheets are attached) Please click here.

Wrap up: Students answer the following question in complete sentences on a sheet of

paper

Explain your learning out of this lesson in a paragraph. (Exit Ticket) Please click here.

Assessment strategies: Lesson quiz will be given to students at the end of the lesson.

(Lesson Quiz is in the PPT.) Please click here.

J. Classroom management: I have three steps for classroom management. First step, I

have personal contact with almost all parents. I am doing home-visits three times a year.

One at the beginning of school year which is in August and September. One in the middle

of the school year which is around in January and the last one is around March. I realized

importance of home-visit when I was in Turkey. I was doing same visits over there too.

By visiting parents home and having personal contact help me to solve my classroom

problems. Second step, seating chart; I have specific seating for every students. I am

preparing this out of my students’ previous teachers’ recommendations. Third step, in our

school we have demerit point system for behavior problems and positive behavior

support system which is for good behaviors. Overall I have very minor problems in my

classrooms.

K. Unit Assessment: I have two quizzes, one test, and one performance assessment for this

unit which are attached.

Summary

Students Name: Kerim Yildiz

Subject of the Project: Geometry - Grade 10 (Parallel and Perpendicular Lines)

Lessons: 4 Lessons which are Lines and Angles, Angels Formed by Parallel Lines and

Transversal, Proving Lines Parallel, and Perpendicular Lines

Rationale: The study of geometry teaches not only geometry but the systematic dissection and

construction of a problem.  This unit codifies knowledge from previous courses of study into a

firm base that will be used to further study geometry.

Standards: G.CO.9. Prove theorems about lines and angles. G.CO.12. Make formal geometric

constructions with a variety of tools and methods.

Goals: Identify parallel, perpendicular, skew lines, and the angles formed by two lines and a

transversal classify pairs of angles. Find angle measures by solving systems of equations. Prove

and use theorems about the angles formed by parallel lines and a transversal and two lines are

parallel using the angles formed by a transversal. Determine whether lines are parallel,

perpendicular or neither. Prove and apply theorems about perpendicular lines

Big Ideas/Essential Understanding: In fields such as architecture and construction, you must

ensure that opposite walls of a building are parallel and adjacent walls are perpendicular

(otherwise your house would fall down).

Essential Questions: How do alternate exterior angles differ from alternate interior angles?

What can you conclude about two coplanar lines that are perpendicular to the same line?