Math 3 - Lesson Title: Using the Coordinate Plane for Proofs
Unit 2: Geometric Proofs (Lesson 3 of 4)
Time Frame: 3-4 Days
Essential Questions:
 What is the relationship between the slopes of parallel lines and of perpendicular lines?
 Given a polygon represented in the coordinate plane, what is its perimeter and area?
 How can geometric relationships be proven through the application of algebraic properties to geometric figures
represented in the coordinate plane?
Targeted Content Standard(s):
Student Friendly Learning Targets
Use coordinates to prove simple geometric theorems algebraically.
 G.GPE.4 Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure defined by
four given points in the coordinate plane is a rectangle; prove or
disprove that the point (1, √3) lies on the circle centered at the origin
and containing the point (0, 2).
 G.GPE.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.GPE.7 Use coordinates to compute perimeters of polygons and
areas of triangles and rectangles, (e.g., using the distance formula).
I can…
 Find the equation of a line parallel to
a given line through a given point.
 Find the equation of a line
perpendicular to a given line
through a given point.
 Use coordinates to show lines are
either parallel or perpendicular.
 Use coordinate geometry, such as
the distance formula, to identify and
prove properties of geometric
figures.
 Determine perimeter and area of a
rectangle, rhombus, and square
given its coordinates.
Targeted Mathematical Practice(s):
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
6 Attend to precision.
7 Look for and make use of structure.
8 Look for an express regularity in repeated reasoning.
Supporting Content Standard(s): (optional)
Purpose of the Lesson:
The overarching emphasis in this unit is for students to use the coordinate plane to verify geometric theorems
previously learned in Math 1 or 2. Students will formalize criteria for parallel and perpendicular lines using the
coordinate plane in segment 2, and then apply these along with the distance formula, to special quadrilaterals as they
investigate and determine the properties of several midpoint quadrilaterals in segments 3 and 4.
Explanation of Rigor: (Fill in those that are appropriate.)
Conceptual:
Students will prove parallel lines have
the same slope, and perpendicular
lines have slopes which are opposite
reciprocals (or whose product is -1).
G.GPE. 5
Procedural:
Students will use coordinates to
compute perimeter and area of
quadrilaterals. G.GPE.7
Vocabulary:
parallel
perpendicular
slope
reciprocal
quadrilateral
rectangle
square
midpoint
diagonal
coordinates
rhombus
parallelogram
Application:
Students will use coordinates to
classify a quadrilateral by its
properties. G.GPE.4
Math 3 - Lesson Title: Using the Coordinate Plane for Proofs
Unit 2: Geometric Proofs (Lesson 3 of 4)
Time Frame: 3-4 Days
Essential Questions:
 What is the relationship between the slopes of parallel lines and of perpendicular lines?
 Given a polygon represented in the coordinate plane, what is its perimeter and area?
 How can geometric relationships be proven through the application of algebraic properties to geometric figures
represented in the coordinate plane?
Pre-Assessment: Parallel and Perpendicular Lines Pre-Assessment (Segment 1)
Formative Assessment(s): Parallel and Perpendicular Lines Activity (Segment 1)
Midpoint Madness (Segments 2 and 3)
Summative Assessment: G.GPE Summative Assessment #1,2,6
Self-Assessment: Self Assessment
Math 3 - Lesson Title: Using the Coordinate Plane for Proofs
Unit 2: Geometric Proofs (Lesson 3 of 4)
Time Frame: 3-4 Days
Essential Questions:
 What is the relationship between the slopes of parallel lines and of perpendicular lines?
 Given a polygon represented in the coordinate plane, what is its perimeter and area?
 How can geometric relationships be proven through the application of algebraic properties to geometric figures
represented in the coordinate plane?
Lesson Procedures:
Segment 1
Approximate Time Frame:
20 minutes
Focus:
Pre-assessing upcoming necessary
skills with linear equations.
Lesson Format:
Resources:
Whole Group
Small Group
Independent
Parallel and Perpendicular Lines PreAssessment
Modeled
Guided
Collaborative
Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
Differentiation for Remediation:
MP#7 Look for and make use of structure. Students
should recognize the usefulness of putting an equation in
slope-intercept form to identify the slope and y-intercept.
Students who do not recognize from the equation the
properties of parallel and perpendicular lines could use a
graphing calculator to investigate the graphs of the
systems.
Differentiation for English Language Learners:
A picture definition of parallel and perpendicular lines
could be provided.
Differentiation for Enrichment:
Potential Pitfall(s):
Independent Practice (Homework):
Students may have trouble getting the equations into
slope intercept form.
Steps:
Teacher Notes/Reflections:
1. Give pre-assessment. As students are completing it, observe student
responses and look for different methods of determining the lines
parallel or perpendicular.
2. Have students share when most are finished. Ask students who solved
the problems using a graph, using only the equations, using points, or
other methods to share their method with the class.
Teacher Notes/Reflections:
Math 3 - Lesson Title: Using the Coordinate Plane for Proofs
Unit 2: Geometric Proofs (Lesson 3 of 4)
Time Frame: 3-4 Days
Essential Questions:
 What is the relationship between the slopes of parallel lines and of perpendicular lines?
 Given a polygon represented in the coordinate plane, what is its perimeter and area?
 How can geometric relationships be proven through the application of algebraic properties to geometric figures
represented in the coordinate plane?
Segment 2
Approximate Time Frame:
Lesson Format:
45 minutes
Whole Group
Small Group
Independent
Focus:
Modeled
Guided
Collaborative
Assessment
Prove the slope criteria for parallel
and perpendicular lines.
Math Practice Look For(s):
MP #2 Reason abstractly and quantitatively. Students will
be able to use figures and information pertaining to a
specific geometric object as an aid in reasoning about that
geometric object in general.
MP #7 Look for and make use of structure. Students will
be able to use the structure of geometric objects to gain
insights into, make conjectures about, and create proofs
pertaining to these objects.
Resources:
Parallel and Perpendicular Lines
Activity
Graph Paper
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Differentiation for Remediation:
Students may use the corner of a notecard to trace a line
on the graph on which the 90⁰ rotation of the original
point would lie. Then, mark off the distance to the original
point from the origin on the side of the notecard. Find the
location of the image by using this marked distance from
the origin on the new line.
Differentiation for English Language Learners:
Differentiation for Enrichment:
Potential Pitfall(s):
Students may not need assistance determining the effect
of a translation or rotation on the coordinates of a point.
Independent Practice (Homework): #1-7 following
Parallel and Perpendicular Lines Activity
Steps:
Teacher Notes/Reflections:
1. In small groups, have students work through #1 in the activity.
Facilitate learning by encouraging students to record how they know
the two lines are parallel with an algebraic method (slope formula).
Wrap up #1 as a class, making sure to reiterate that parallel lines can
be created by a sequence of two translations.
Students may be interested in investigating
any additional transformations that
produce parallel lines.
2. Have students continue working on #2-5 of the activity. Encourage
students to use algebraic formulas to calculate the slope, instead of
just counting the rise and run on the graph. Observe how the
students explain the lines are perpendicular using slope. If different
arguments surface, have a whole class discussion about the
similarities of their ideas, (e.g., the product of the slopes is -1, the
slopes are opposite reciprocals).
Teacher Notes/Reflections:
Math 3 - Lesson Title: Using the Coordinate Plane for Proofs
Unit 2: Geometric Proofs (Lesson 3 of 4)
Time Frame: 3-4 Days
Essential Questions:
 What is the relationship between the slopes of parallel lines and of perpendicular lines?
 Given a polygon represented in the coordinate plane, what is its perimeter and area?
 How can geometric relationships be proven through the application of algebraic properties to geometric figures
represented in the coordinate plane?
Segment 3
Approximate Time Frame:
45 minutes
Focus:
Use coordinates to prove simple
geometric theorems algebraically,
specifically focused on definitions of
quadrilaterals.
Lesson Format:
Resources:
Whole Group
Small Group
Independent
Midpoint Madness
Graph Paper
Modeled
Guided
Collaborative
Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Use the slope criteria for parallel and
perpendicular lines to solve geometric
problems.
Math Practice Look For(s):
MP #2 Reason abstractly and quantitatively. Students will
be able to use figures and information pertaining to a
specific geometric object as an aid in reasoning about that
geometric object in general.
MP #7 Look for and make use of structure. Students will
be able to use the structure of geometric objects to gain
insights into, make conjectures about, and create proofs
pertaining to these objects.
MP #6 Attend to precision. Students will recognize that
incorrect initial attempts at definitions, conjectures, and
theorems may be corrected through a process of
refinement.
MP #8 Look for and express regularity in repeated
reasoning. Students will recognize a pattern in the
classification of the midpoint quadrilaterals and generalize
a pattern in the area and perimeter of these
quadrilaterals.
Differentiation for Remediation:
Potential Pitfall(s): Students who do not understand the
similarities and differences between the types of
quadrilaterals may need some remediation.
Independent Practice (Homework): Assign #14 for
homework. Sketches are fine, but students could use
graph paper to be more precise.
Students may need to list the properties of quadrilaterals
and be guided to see the relationships between all the
types. Dynamic geometry software may be helpful to
demonstrate these connections.
Differentiation for English Language Learners:
When stating properties of quadrilaterals, students may
draw a picture to support their reasoning.
Differentiation for Enrichment:
Steps:
Teacher Notes/Reflections:
1. Use whole class instruction to guide students in #1-6.The quadrilateral
ABCD is the solution to the homework from Segment 2. Students may
need a refresher on the properties of parallelograms, rectangles,
rhombi, and squares.
2. Allow students to work in small groups for #7-13. Encourage students
to state geometric properties using If-Then statements when
Teacher Notes/Reflections: Students may
need guidance when placing the notecard
Math 3 - Lesson Title: Using the Coordinate Plane for Proofs
Unit 2: Geometric Proofs (Lesson 3 of 4)
Time Frame: 3-4 Days
Essential Questions:
 What is the relationship between the slopes of parallel lines and of perpendicular lines?
 Given a polygon represented in the coordinate plane, what is its perimeter and area?
 How can geometric relationships be proven through the application of algebraic properties to geometric figures
represented in the coordinate plane?
supporting their answers. Possible observations might include:
on the coordinate plane, as this can be
1) If a quadrilateral has four congruent sides, then it is a rhombus.
done abstractly, naming the coordinates
2) If the diagonals of a quadrilateral bisect each other and are
by the measurements of the notecard, i.e.
perpendicular, then it is a rhombus.
(0,4)(6,4)(0,0)(6,0).
3) If a quadrilateral has 2 pairs of opposite sides congruent, then it is
a parallelogram.
4) If a parallelogram has one right angle, then it is a rectangle.
3. When most groups have completed #9, it may be helpful to share
different reasoning statements in a whole group.
Teacher Notes/Reflections:
4. To complete #12, students should be using the distance formula.
Teacher Notes/Reflections: In #11 the
1
pattern in table the area is 24  
2
1
for perimeter 20  
2
n
1
n1
n1
n is odd and
 1 2
4 13   if n is even.
2
and
Math 3 - Lesson Title: Using the Coordinate Plane for Proofs
Unit 2: Geometric Proofs (Lesson 3 of 4)
Time Frame: 3-4 Days
Essential Questions:
 What is the relationship between the slopes of parallel lines and of perpendicular lines?
 Given a polygon represented in the coordinate plane, what is its perimeter and area?
 How can geometric relationships be proven through the application of algebraic properties to geometric figures
represented in the coordinate plane?
Segment 4
Approximate Time Frame:
30 minutes
Focus:
Use coordinates to prove simple
geometric theorems algebraically,
specifically focused on definitions of
quadrilaterals.
Lesson Format:
Resources:
Whole Group
Small Group
Independent
Midpoint Madness
Graph Paper
Modeled
Guided
Collaborative
Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Use the slope criteria for parallel and
perpendicular lines to solve geometric
problems.
Math Practice Look For(s):
Differentiation for Remediation:
MP #3 Construct viable arguments and critique the
reasoning of others. Students will be able to create and
present a proof that the midpoint quadrilateral of a
quadrilateral is a rectangle, and be able to critique the
proof and deductive reasoning of others.
Students who have difficulty in the abstract case may use
numerical values for the coordinates and complete the
activity for several different quadrilaterals.
MP #7 Look for and make use of structure. Students will
be able to use the structure of geometric objects to gain
insights into, make conjectures about, and create proofs
pertaining to these objects.
Potential Pitfall(s):
Differentiation for English Language Learners:
Differentiation for Enrichment:
Independent Practice (Homework):
Students may have difficulty generalizing and may need to
use numerical values at first.
Steps:
1. Students should work in small groups to devise a proof and support their
reasoning using the distance formula and slope criteria for parallel and
perpendicular lines.
Teacher Notes/Reflections: