8th Grade Geometry MSG1 Unit Plan
Day 1
Parallel Lines
Day 2
Parallel Lines Continued
Day 3
Perpendicular Lines
Day 4
Perpendicular Lines Continued
Day 5
Transversals that Cut Parallel Lines
(Angle Properties)
Day 6
Transversals that Cut Parallel Lines
(Angle Properties) Continued
Day 7
Ratios of Segments of Parallel Lines Cut by
One or More Transversals
Day 8
Congruency of Geometric Figures
Day 9
End of Unit Task
Day 10
Presentations
Eighth Grade Geometry
Cynthia Thomas
Page 1 of 12
Parallel Lines
Days 1-2
Time: 1.5 hours (1 block period or 2 -50 minute periods)
Standard:
M8G1. Students will understand and apply the properties of parallel and perpendicular lines
and understand the meaning of congruence.
a. Investigate characteristics of parallel and perpendicular lines both algebraically and
geometrically.
Materials:
-projector screen with Geometer’s Sketchpad (for the GSP files for constructions), if these are not
available the teacher can do the constructions manually on the board.
-GSP files (included)
-graph paper, a sheet is attached that can be copied for the students if needed
-compass (1 per student)
-ruler (1 per student)
-pictures of Euclid and Descartes (attached), to add interest to the history portions of the lesson
Previous Knowledge:
▪ Students have studied constructions of parallel lines in seventh grade,
M7G1. Students will construct plane figures that meet given conditions.
a. Perform basic constructions using both compass and straight edge, and appropriate
technology. Constructions should include 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.
The lesson includes a review of constructing parallel lines but depending on the students it may take
more/less time.
▪ Students are required to rearrange equations into slope-intercept form in this lesson.
▪ Graphing skills (students observe and make conjectures about parallel lines on coordinate plane).
Lesson Procedure
1. Introduction: 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!
2. 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.
Eighth Grade Geometry
Cynthia Thomas
Page 2 of 12
3. Constructing parallel and perpendicular lines using only a ruler and compass. The attached
GSP sketch – Parallel - takes the students (and teacher) through this process step-by-step.
Note: This should be a review from seventh grade. Hence, the constructions should not
take up the entire class period.
4. Students should first construct a line parallel to a given line through a given point. They
should follow along with the GSP sketch in order to remember the construction.
5. After discussion of the construction process (if students have questions), students should
practice alone or in pairs with constructing parallel lines on their own. The teacher should
monitor their progress.
6. Connecting to Algebra. History: René Descartes. **Photo of Descartes attached at end of
lesson.
René Descartes was a 17th Century mathematician, philosopher, and physicist. One of his
most noted achievements in mathematics is Cartesian Geometry. Cartesian geometry is the
study of geometry using the coordinate plane and algebra, a different approach from Euclidean
geometry, where nothing is done on the coordinate plane.
We are going to do some Cartesian geometry today!
7. Connecting to Algebra:
▪ How do you know if two lines are parallel without actually constructing the lines?
▪ Can you tell just by looking at the equations of the two lines?
Let’s test these questions by using a coordinate plane.
Directions to students:
a. Construct two parallel lines (using the method we have reviewed today) on graph
paper.
b. In pairs/groups or individually, write down everything you observe about the parallel
lines in relation to the graph.
(It might be helpful to write a few prompts on the board for the students, such as:
-If you draw an x and y-axis, where are the intercepts?
-Can you determine the slopes of the two lines?)
8. Discuss what the students noticed from constructing the parallel lines on the graph. The
intention is for them to notice the lines have the same slope.
9. If parallel lines have the same slope then two lines can be determined to be parallel by
looking at their equations. Why?
Further Discussion: what makes two lines parallel? How can parallelism be looked at in
an algebraic manner? Parallel does not determine the distance between lines but what
does have to be the same for lines to be parallel?
10. Review: slope-intercept form of equations.
Students should put these two equations in slope-intercept form:
a. y – 9 = 3x
b. -3x = -y – 7
11. Once students have the equations in slope-intercept form (this may require a review of what
slope-intercept form is), they need to make observations about the equations. What
similarities and differences do the two equations have?
12. Notice their slopes are exactly the same (3).
13. Quick activity!
b. Have students get into pairs.
c. Each student need to come up with equations of two lines that are parallel.
Students may choose to write the equations in slope-intercept form to better
ensure their two lines will be parallel. Then they should rearrange the
equations so they are not in slope-intercept form.
Eighth Grade Geometry
Cynthia Thomas
Page 3 of 12
d. Students switch papers so their partner has their two equations and vice versa.
e. Each student must first write the two equations in slope-intercept form and then
graph the two equations on graph paper (can be provided).
14. Wrap-up
Students answer the following question in complete sentences on a sheet of paper:
What are two methods of determining whether or not lines are parallel?
Next Lesson:
Perpendicular Lines
Eighth Grade Geometry
Cynthia Thomas
Page 4 of 12
Photos for history in Parallel Lines Lesson
Euclid
Descartes
Eighth Grade Geometry
Cynthia Thomas
Page 5 of 12
Perpendicular Lines
Days 3-4
Time: 1.5 hours (1 block period or 2 -50 minute periods)
Standard:
M8G1. Students will understand and apply the properties of parallel and perpendicular lines
and understand the meaning of congruence.
a. Investigate characteristics of parallel and perpendicular lines both algebraically and
geometrically.
Materials:
-projector screen with Geometer’s Sketchpad (for the GSP files for constructions), if these are not
available the teacher can do the constructions manually on the board.
-GSP files (included)
-graph paper, a sheet is attached that can be copied for the students if needed
-compass (1 per student)
-ruler (1 per student)
-pictures of Euclid and Descartes (attached), to add interest to the history portions of the lesson
Previous Knowledge:
▪ Students have studied constructions of perpendicular lines in seventh grade,
M7G1. Students will construct plane figures that meet given conditions.
f. Perform basic constructions using both compass and straight edge, and appropriate
technology. Constructions should include 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.
The lesson includes a review of constructing perpendicular lines but depending on the students it
may take more/less time.
▪ Students must understand what right angles are for this lesson.
▪ Students are required to rearrange equations into slope-intercept form in this lesson.
▪ Graphing skills (students observe and make conjectures about perpendicular lines on coordinate
plane).
▪ It may be necessary to introduce the idea of a “reciprocal” number to students prior to or during
this lesson, if they have not encountered that term/concept before.
Lesson Procedure
1. Discussion: What does it mean for two lines to be perpendicular?
a. Lines that intersect at right angles.
b. ***This would be an appropriate time to review what a right angle is in order to
refresh students’ memories.
2. Constructing parallel and perpendicular lines using only a ruler and compass. The attached
GSP sketch – Perpendicular - takes the students (and teacher) through this process step-bystep.
Note: This should be a review from seventh grade. Hence, the constructions should not
take up the entire class period.
3. Students should first construct a line parallel to a given line through a given point. They
should follow along with the GSP sketch in order to remember the construction.
Eighth Grade Geometry
Cynthia Thomas
Page 6 of 12
4. After discussion of the construction process (if students have questions), students should
practice alone or in pairs with constructing perpendicular lines on their own. The teacher
should monitor their progress.
5. Connecting to Algebra:
▪ How do you know if two lines are perpendicular without actually constructing the
lines?
▪ Can you tell just by looking at the equations of the two lines?
Let’s test these questions by using a coordinate plane.
Directions to students:
a. Construct two perpendicular lines (using the method we have reviewed today) on
graph paper.
b. In pairs/groups or individually, write down everything you observe about the
perpendicular lines in relation to the graph.
(It might be helpful to write a few prompts on the board for the students, such as:
-If you draw an x and y-axis, where are the intercepts?
-Can you determine the slopes of the two lines?)
6. Discuss what the students noticed from constructing the perpendicular lines on the graph.
The intention is for them to notice the lines have the opposite reciprocal slopes.
7. If perpendicular lines have opposite reciprocal slopes then two lines can be determined to be
perpendicular by looking at their equations. Why?
Further Discussion: What makes two lines perpendicular? How can perpendicularity be
looked at in an algebraic manner?
8. Review: slope-intercept form of equations.
Students should put these two equations in slope-intercept form:
a. y – 9 = 3x
b. (1/3)x = -y – 7
9. Once students have the equations in slope-intercept form, they need to make observations
about the equations. What similarities and differences do the two equations have?
10. Notice their slopes are opposite reciprocals. This means the slope of one is the reciprocal
times -1 of the slope of the other equation, 3 and -1/3.
11. Quick activity!
g. Have students get into pairs.
h. Each student need to come up with equations of two lines that are perpendicular.
Students may choose to write the equations in slope-intercept form to better
ensure their two lines will be perpendicular. Then they should rearrange the
equations so they are not in slope-intercept form.
i. Students switch papers so their partner has their two equations and vice versa.
j. Each student must first write the two equations in slope-intercept form and then
graph the two equations on graph paper (can be provided).
12. Connection between parallel and perpendicular lines.
Have students graph two parallel lines and a line perpendicular to one of the parallel lines.
What do you notice?
-The perpendicular line is perpendicular to BOTH parallel lines.
***If time allows, let students explore this idea in partners as done before with parallel and
perpendicular lines.
13. Wrap-up
Students answer the following question in complete sentences on a sheet of paper:
What are two methods of determining whether or not lines are perpendicular?
Next Lesson:
Transversals
Eighth Grade Geometry
Cynthia Thomas
Page 7 of 12
Parallel Lines Cut by a Transversal
Days 5-6
Time: 1.5 hours (1 block period or 2 -50 minute periods)
Standard:
M8G1. Students will understand and apply the properties of parallel and perpendicular lines
and understand the meaning of congruence.
b. Apply properties of angle pairs formed by parallel lines cut by a transversal.
Materials:
-projector screen with Geometer’s Sketchpad, if these are not available the teacher can draw the
angles manually on the board.
-GSP files (included)
-graph paper, a sheet is attached that can be copied for the students if needed
-compass (1 per student)
-ruler (1 per student)
-pictures of Euclid and Descartes (attached), to add interest to the history portions of the lesson
Previous Knowledge:
▪ Students should understand what an angle is and how an angle is measured in degrees.
▪ Students should understand the previous two lessons on parallel and perpendicular lines.
Lesson Procedure
1. Tell the students that today we will be looking at parallel lines when they are cut by a
transversal.
2. Use GSP Sketch – Transversals – for the lesson.
3. After you have gone through the GSP Sketch with the class (it may take an entire class period or
more), it is important to show students how we notate angles with equal measure in
mathematics. This takes just a few minutes. (Ex: Draw angles and show how you draw an arc
across the angle and then place the same symbol, such as a dash, on both arcs of the congruent
angles).
4. 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!
5. It also important to discuss the converse of each of the theorems presented in the GSP Sketch.
Meaning that if two alternate interior angles are congruent then the lines (aside from the
transversal) must be parallel.
Next Lesson
Ratio of Segments of Transversals that cut Parallel Lines
Eighth Grade Geometry
Cynthia Thomas
Page 8 of 12
Ratio of Segments of Transversals that Cut Parallel Lines
Day 7
Time: 50 minutes
Standard:
M8G1. Students will understand and apply the properties of parallel and perpendicular lines
and understand the meaning of congruence.
c. Understand the properties of the ratio of segments of parallel lines cut by one or more transversals.
Materials:
-projector screen with Geometer’s Sketchpad, if these are not available the teacher can draw the
angles manually on the board.
-GSP files (included)
-graph paper, a sheet is attached that can be copied for the students if needed
-compass (1 per student)
-ruler (1 per student)
-pictures of Euclid and Descartes (attached), to add interest to the history portions of the lesson
Previous Knowledge:
▪ Students should understand what ratio is.
▪ Students should understand the previous 3 lessons on parallel and perpendicular lines and
transversals.
Lesson Procedure
1. Review what a transversal is. Have students name a few angle pairs that can be located when a
transversal cuts two parallel lines.
2. Show GSP Sketch – Ratio.
3. Quick Activity.
a. Have students get into pairs.
b. Students should use their compasses and rulers to draw three parallel lines and then use
their rulers to draw two transversals that cut the three parallel lines.
c. Using what they have just learned, they should identify any ratios that are present.
d. Using their rulers, and measuring to the nearest millimeter, they should measure to
verify the ratios they identified.
Next Lesson
Congruence
Eighth Grade Geometry
Cynthia Thomas
Page 9 of 12
Congruence
Day 8
Time: 50 minutes
Standard:
M8G1. Students will understand and apply the properties of parallel and perpendicular lines
and understand the meaning of congruence.
c. Understand the meaning of congruence: that all corresponding angles are congruent and all
corresponding sides are congruent
Materials:
-projector screen with Geometer’s Sketchpad, if these are not available the teacher can draw the
angles manually on the board.
-GSP files (included)
-compass (1 per student)
-ruler (1 per student)
Previous Knowledge:
▪ Students should understand what corresponding sides and corresponding angles are when dealing
with geometric shapes.
Lesson Procedure:
1. Review the meaning of congruence. Congruence means equal. When talking about sides it
means they have the same length. When talking about angles it means they have the same
measure.
2. Show GSP Sketch – Congruence.
3. Worksheet – Are These Congruent?.
4. If students do not complete the worksheet in class they should finish it for homework.
Next Lesson
End of Unit Task
Eighth Grade Geometry
Cynthia Thomas
Page 10 of 12
End of Unit Task
Day 9
Time: 50 minutes
Standard:
M8G1. Students will understand and apply the properties of parallel and perpendicular lines
and understand the meaning of congruence.
Materials:
-Computers with presentation software
Or
-Poster Paper and drawing materials
Previous Knowledge:
Previous 8 days of lessons.
Task:
1. The point of this activity is for students to take a day to organize everything they have just
learned. Students should be broken into 5-6 groups.
2. Each group will be assigned a topic from the unit, a possible way to split the topics:
1. Parallel Lines
2. Perpendicular Lines
3. Transversals (angle properties)
4. Ratios of Segments of Transversals
5. Congruency
(The topics can be split however the teacher wants. If it is preferred to be a partner
project some pairs may do the same topics).
3. Ideally students can create some way to present their topic to the class. This can be done
through PowerPoint or other presentation software, Geometer’s SketchPad, or simply on poster
paper with markers.
4. Have the students brainstorm about how to present their topic and then use their notes from class
to create their presentations.
Next Day
Presentations
Eighth Grade Geometry
Cynthia Thomas
Page 11 of 12
End of Unit Task - Presentations
Day 10
Time: 50 minutes
Standard:
M8G1. Students will understand and apply the properties of parallel and perpendicular lines
and understand the meaning of congruence.
Materials:
-Computers with presentation software
Or
-Poster Paper and drawing materials
Previous Knowledge:
Previous 8 days of lessons and 9th day creating presentation.
Presentations
The students should present their topics to the class. Remind the class that this is a great
opportunity to make their notes clearer on all the topics. Students should remember to be respectful
during the presentations.
Eighth Grade Geometry
Cynthia Thomas
Page 12 of 12