DRAFT
Module Guidance Document
Geometry-R Module 4
Topic A
Topic B
Topic C
Topic D
Rectangular and Triangular
Regions Defined By
Inequalities
Perpendicular and Parallel
Lines in the Cartesian plane
Perimeters and Areas of
Polygonal Regions in the
Cartesian Plane
Partitioning and Extending
Segments and
Parameterization of Lines
4 days
6 days
5 Days
5 days
OVERVIEW
In this module, students explore and experience the utility of analyzing algebra and geometry challenges through the framework of coordinates.
To set the stage for complex work in analytic geometry (computing coordinates of points of intersection of lines and line segments or the
coordinates of points that divide given segments in specific length ratios, and so on), students will describe the region via systems of algebraic
inequalities (A-REI.D.12).
To fully develop the analysis of perimeter and area of a polygon in terms of the coordinates of its vertices (G-GPE.B.7), students will derive the
area 𝐴 of a triangle with coordinates (0,0), (𝑥1,𝑦2), and (𝑥2,𝑦2) as 𝐴=1/2|𝑥1𝑦1−𝑥2𝑦2| and extend this result to the areas of triangles situated
elsewhere in the plane and to simple polygons seen as unions of triangles.
Students will also find locations on a directed line segment between two given points that partition the segment in given ratios (G-GPE.B.6) and
connect this work to proving classical results in geometry (G-GPE.B.4). For example, proving that the diagonals of a parallelogram bisect one
another, and the medians of a triangle meet at the point 2/3 of the way from the vertex for each.
DRAFT
Focus Standards
Module Guidance Document
Use coordinates to prove simple geometric theorems algebraically.2
G-GPE.B.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.B.5 Prove3 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.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★
Focus Standards for Mathematical Practice
MP.2 Reason abstractly and quantitatively. Students rotate line segments about their endpoints and discover the general slope criterion for
perpendicular lines and articulate this criterion in an abstract setting. Geometric results (such as “the three medians of a triangle are
concurrent”) are examined in concrete settings and students determine that these results hold in general. They also develop a formula for the
area of a triangle based solely on the coordinates of its three vertices and generalize this to an area formula for quadrilaterals and other planar
polygons.
MP.7 Look for and make use of structure. Students determine slope criteria for perpendicular and parallel lines and use these slope conditions
to develop the general equation of a line and the formula for the distance of a point from a line. Students determine the area of polygonal
regions using multiple methods including Green’s theorem and decomposition. Definitive geometric properties of special quadrilaterals are
explored and properties of special lines in triangles are examined.
MP.8 Look for and express regularity in repeated reasoning. Students use the midpoint to repeatedly separate a segment into proportional
parts and derive a formula for calculating the coordinates of a point that will divide a segment into segments of given ratios.
DRAFT
Lesson
Big Idea
Module Guidance Document
Emphasize
Suggested
Problems
(Problem Set)
Exit Ticket
Suggested
Days
TOPIC A
1
2
SKIP
Finding Systems of Inequalities
That Describe Triangular and
Rectangular Regions
Opening Exercises 1-2,
Example 1, Exercise 1, 3,
Example 2, Exercise 4
2, 4, 5
Yes
2
3
Yes
2
Students describe rectangles (with
edges parallel to the axes) and
triangles in the coordinate plane by
means of inequalities. For example,
the rectangle in the coordinate plane
with lower left vertex (1,2) and
upper right vertex (10,15) is
{(𝑥,𝑦)│1≤𝑥≤10 & 2≤𝑦≤15}; the
triangle with vertices at (0,0), (1,3),
and (2,1) is {(𝑥,𝑦)│𝑥/2≤𝑦≤3𝑥 &
𝑦≤−2𝑥+5}.
3
Lines That Pass Through
Regions
Given two points in the coordinate
plane and a rectangular or triangular
region, students determine whether
the line through those points meets
the region, and if it does, they
describe the intersections as a
segment and name the coordinates
of the endpoints.
4
SKIP
Opening Exercise, Example 1,
Example 2 a, b, c
DRAFT
Lesson
Big Idea
Module Guidance Document
Emphasize
Suggested
Problems
(Problem Set)
Exit Ticket
Suggested
Days
TOPIC B
5
Criterion for Perpendicularity

Students explain the connection
between the Pythagorean theorem
and the criterion for
perpendicularity.
Opening Exercise, Example 1,
Exercise 1, Example 2
(Theorem-IMPORTANT),
Exercises 1-3, Example 3
(Theorem-IMPORTANT)
1, 2, 4, 7
Yes
2
No
1
No
1
Additional Practice
Problems
Holt Pg 195 33-44
Jmap Parallel and
perpendicular
Jmap Parallel and
perpendicular
Jmap Parallel and
Perpendicular
6
Segments That Meet at Right
Angles

Opening Exercise
Problem Set 1,3,4, 5
Students generalize the criterion for
perpendicularity of two segments
that meet at a point to any two
segments in the Cartesian plane.
Students apply the criterion to
determine if two segments are
perpendicular.
7
Equations for Lines Using
Normal Segments
Students state the relationship
Opening exercise, Example 1,
Exercises 1-2
Holt Resources
Problem Set 1,2,4
DRAFT
Lesson
Big Idea
Module Guidance Document
Emphasize
Suggested
Problems
(Problem Set)
Exercise 1 a, b, c, Exercises 15,
Problem Set 1,2
Exit Ticket
Suggested
Days
between previously used formats for
equations for lines and the new
format 𝑎1𝑥+𝑎2𝑦=𝑐, recognizing the
segments from (0,0) to (𝑎1,𝑎2) as a
normal and −𝑎2/𝑎1 as a slope.
8
Parallel and Perpendicular
Lines
*Students recognize parallel and
perpendicular lines from slope.
Students create equations for
lines satisfying criteria of the kind:
“Contains a given point and is
parallel/perpendicular to a given
line.”
Suggested Problems from Mid Module Assessment
1, 2, 4, 5, 7, 9, 10
TOPIC C
yes
2
DRAFT
Lesson
9
Big Idea
Perimeter and Area of
Triangles in the Cartesian
Plane
Module Guidance Document
Emphasize
Opening Exercises, Example
1, Exercise 1, Example 2
(IMPORTANT)
Suggested
Problems
(Problem Set)
Exit Ticket
Suggested
Days
No
2
No
1
YES
2
Problem Set 1-3
JMap Triangles in the
coordinate plane

Students find the perimeter of a
triangle in the coordinate plane
using the distance formula.
Students state and apply the
formula for area of a triangle with
vertices (0,0), (x1,𝑦1), and (𝑥2,𝑦2).
10
Perimeter and Area of
Polygonal Regions in the
Cartesian Plane
Opening exercise, Exercises 1,
2, Discussion
Jmap Quads in the
Coordinate Plane
Jmap Quads in the
plane Part 2

Students find the perimeter of a
quadrilateral in the coordinate plane
given its vertices and edges.
Students find the area of a
quadrilateral in the coordinate plane
given its vertices and edges by
employing Green’s theorem.
11
Perimeters and Areas of
Polygonal Regions Defined by
Systems of Inequalities

Students find the perimeter of a
Opening Exercises, Exercises
1-2
Problem Set, 1-5
DRAFT
Lesson
Big Idea
Module Guidance Document
Emphasize
Suggested
Problems
(Problem Set)
Exit Ticket
Suggested
Days
triangle or quadrilateral in the
coordinate plane given a description
by inequalities.
Topic D
12
Dividing Segments
Proportionately
Opening Exercise, Example 1,
Example 2, Exercises 1-3
Problem Set 1-3
Yes
2
No
1
Students find midpoints of segments
and points that divide segments into
3, 4, or more proportional, equal
parts.
13
Analytic Proofs of Theorems
Previously Proved by
Synthetic Means
Using coordinates, students
prove that the intersection of the
medians of a triangle meet at a point
that is two-thirds of the way along
each median from the intersected
vertex.
Using coordinates, students
prove the diagonals of a
parallelogram bisect one another
and meet at the intersection of the
segments joining the midpoints of
opposite sides.
14
SKIP
15
SKIP
Suggested problems from End Mod Assessment
Opening, exercise 1(Median
Concurrence), Exercise 2
(parallelograms have
diagonals that bisect each
other)
DRAFT
Lesson
Big Idea
Module Guidance Document
Emphasize
1, 2, 5, 6
Holt Additional Practice Problems
Holt Pg 195 47-52
Is this A Rectangle? from Illustrativemathematics.org
Criterion for Perpendicular Lines from illustrativemathematics.org
Finding Triangle Coordinates from illustrativemathematics.org
Suggested
Problems
(Problem Set)
Exit Ticket
Suggested
Days
DRAFT
Problems from NYSED Released Sample Items
Module Guidance Document
DRAFT
Module Guidance Document