Lesson Title: Proving Slope Criteria for Parallel & Perpendicular Lines Course: CC Geometry, Unit 1
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
G.GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a
given point).
G.CO.A.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.
This lesson will engage students in the following mathematical practices:
MP2: Reason abstractly and quantitatively.
MP6: Attend to precision.
MP8: Look for and express regularity in repeated reasoning.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Assess students’ prior knowledge of slopes and the
properties of rectangles using these questions.
1. What is true about the opposite sides in any
rectangle?
2. Plot the following three points on graph paper,
labeling and connecting them with segments as
you go: A (3,4), B (0,0), and C (6, -4).
3. Find the coordinates of a fourth point D that
would create Rectangle ABCD when segments
are drawn to it from the other points plotted.
4. Find the slope of one side of the rectangle.
5. Find the length of one side of the rectangle
using the distance formula.
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Have the students complete a summary sheet for parallel
and perpendicular lines, listing everything they know
about each, including the slope criteria. For example,
Parallel Lines
 Equidistant apart (do
not intersect)
 Equal slopes
 No solution in a system
of equations
Perpendicular Lines
 Intersect at right angles
 Product of slopes is -1
(or slopes are negative
reciprocals)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Divide students into small groups of 3- 4 students to work at stations.
2. Create four stations for investigating the slope of parallel and perpendicular lines. You may have to set
up more than one station of each type, depending on class size. (See the Parallel and Perp. Lines
Stations document.) Look for evidence of math practices (MP) indicated while students work.
 Station A- Students will use the distance formula to verify lines are parallel by finding the
constant distance between them. Then they will find the slopes of the lines. Look for evidence of
MP6 as students calculate distances and find slopes. (Use rectangles drawn in a coordinate plane
for the investigation. See Station A activity sheet.)
 Station B- Students will use Patty Paper to investigate slopes of parallel and perpendicular lines.
Tell students, “Be sure to discuss patterns you see after finding the slopes of these lines.” (MP8)
 Station C- Students will investigate slopes of parallel and perpendicular lines using Geometers’
Sketchpad. Have one pair in each group work as partners on Part I and the other pair on Part II.
Look for evidence of MP2 as students analyze measurements and make conjectures at this station.
(Note: Set up this activity in advance. Create the sketches for Parts I and II beforehand by doing
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Proving Slope Criteria for Parallel & Perpendicular Lines Course: CC Geometry, Unit 1
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
and saving Steps 1-3 as a file, and have the students do the remaining steps at this station.)
 Station D- Students will investigate slopes of parallel and perpendicular lines formed by similar
right triangles that share a common vertex. Look for evidence of MP6 as students find measures.
3. Allow students to have about 5 minutes per station. Each member of the group will do a different
problem so that there is only one problem per student completed at each station. Once students have
rotated through all four stations, have them work in their small groups to discuss their findings and
write a conjecture about the slope criteria for parallel and perpendicular lines.
4. Conduct a class discussion about the results of the investigations students completed at stations.
5. Verify the criteria for slopes of parallel and perpendicular lines by giving students the formal rules.
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
By the end of the lesson, students should be able to state that the slope criteria are (1) for parallel lines, the
slopes are equal and (2) for perpendicular lines, the product of the slopes is -1 or the slopes are negative
reciprocals. They should be able to use the slope criteria to solve geometric problems (e.g., find the
equation of a line parallel or perpendicular to a given line that passes through a given point).
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Vocabulary: Parallel, Perpendicular, Slope, Distance, Rectangle, Right Triangle, Opposite sides
Note: This lesson may take more than one day.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Graph paper
Straightedge
“Parallel and Perp. Lines Stations” Worksheets
Computer or Tablets for Sketchpad Investigation
Patty Paper
For homework, assign problems involving
application of the slope criteria for parallel and
perpendicular lines (e.g., have them find the equation
of a line parallel or perpendicular to a given line that
passes through a given point and use the criteria in
real world problems).
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
How well did these explorations help develop students’ conceptual understanding of the slope criteria for
parallel and perpendicular lines?
Are students able to state and apply the criteria accurately?
Are students able to identify the criteria when presented in different applications?
Can all students apply the criteria in problem-solving situations, i.e. related to writing the equation of a line
or solving systems of equations?
What real world applications or extensions to learning can I introduce to help students make meaningful
connections for this concept?
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.