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Standard #: MAFS.912.G-GPE.2.5
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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).
Subject Area: Mathematics
Grade: 912
Domain: Geometry: Expressing Geometric Properties with Equations
Cluster: Use coordinates to prove simple geometric theorems algebraically. (Geometry - Major
Cluster) Clusters should not be sorted from Major to Supporting and then taught in that order. To do so
would strip the coherence of the mathematical ideas and miss the opportunity to enhance the
major work of the grade with the supporting clusters.
Date Adopted or Revised: 02/14
Content Complexity Rating: Level 2: Basic Application of Skills & Concepts - More
Information
Date of Last Rating: 02/14
Status: State Board Approved
Remarks/Examples
Geometry - Fluency Recommendations
Fluency with the use of coordinates to establish geometric results, calculate length and angle, and
use geometric representations as a modeling tool are some of the most valuable tools in
mathematics and related fields.
Related Courses
Course Number
1200400:
1207310:
1206310:
1206320:
1206315:
1200410:
7912065:
1200700:
Course Title
Intensive Mathematics (Specifically in versions: 2014 - 2015,
2015 and beyond (current))
Liberal Arts Mathematics (Specifically in versions: 2014 - 2015,
2015 and beyond (current))
Geometry (Specifically in versions: 2014 - 2015, 2015 and
beyond (current))
Geometry Honors (Specifically in versions: 2014 - 2015, 2015
and beyond (current))
Geometry for Credit Recovery (Specifically in versions: 2014 2015, 2015 and beyond (current))
Mathematics for College Success (Specifically in versions: 2014
- 2015, 2015 and beyond (current))
Access Geometry (Specifically in versions: 2015 and beyond
(current))
Mathematics for College Readiness (Specifically in versions:
2014 - 2015, 2015 and beyond (current))
Related Access Points
Access Point
Access Point Number
MAFS.912.G-GPE.2.AP.5a
MAFS.912.G-GPE.2.AP.5b
Access Point Title
Using slope, prove lines are parallel or perpendicular.
Find equations of lines based on certain slope criteria such as;
finding the equation of a line parallel or perpendicular to a given
line that passes through a given point.
Related Resources
Lesson Plan
Name
"When will we ever meet?"
Description
Students will be guided through the investigation of y = mx+b.
Through this lesson, students will be able to determine whether
lines are parallel, perpendicular, or neither by looking at the
graph and the equation.
Starting with a set of three points, students will practice finding
equations of lines and the lines that are perpendicular to them.
Forget Waldo - Where is 'the The students will repeat this process three times - using
orthocenter'?
different colors for differentiating one line from the next. The
big finale brings all the work together and the students realize
this activity leads to finding the orthocenter of a triangle.
The lesson teaches students about an important characteristic of
lines: their slope. Slope can be determined either in graphical or
algebraic form. Slope can also be described as positive,
negative, zero, or undefined. Students get an explanation of
when and how these different types of slope occur. Finally,
Graphing Equations on the
students learn how slope relates to parallel and perpendicular
Cartesian Plane: Slope
lines. When two lines are parallel, they have the same slope and
when they are perpendicular their slopes are negative
reciprocals of one another. Prerequisite knowledge: Students
must know how to graph points on the Cartesian plane. They
must be familiar with the x- and y- axes on the plane in both the
positive and negative directions.
Students will investigate lines to discover the relationships of
Investigating Lines With Our
the slopes of parallel and perpendicular lines using X-Y
Minds!
Coordinate Pegboards.
Problem-Solving Task
Name
Description
This problem solving task gives students the opportunity to
prove a fact about quadrilaterals: that if we join the midpoints of
A Midpoint Miracle
an arbitrary quadrilateral to form a new quadrilateral, then the
new quadrilateral is a parallelogram, even if the original
quadrilateral was not.
This problem solving task challenges students to use ideas about
Triangles inscribed in a circle linear functions in order to determine when certain angles are
right angles.
This problem solving task asks students to find the area of a
Unit Squares and Triangles
triangle by using unit squares and line segments.
Formative Assessment
Name
Description
Finding Equations of Parallel This lesson is intended to help you assess how well students are
and Perpendicular Lines
able to understand the relationship between the slopes of
parallel and perpendicular lines and, in particular, to help
identify students who find it difficult to:


Find, from their equations, lines that are parallel and
perpendicular.
Identify and use intercepts.
It also aims to encourage discussion on some common
misconceptions about equations of lines.
Proving Slope Criterion for
Students are asked to prove that two parallel lines have equal
Parallel Lines - One
slopes.
Proving Slope Criterion for
Students are asked to prove that two lines with equal slopes are
Parallel Lines - Two
parallel.
Proving Slope Criterion for
Students are asked to prove that the slopes of two perpendicular
Perpendicular Lines - 1
lines are both opposite and reciprocal.
Proving Slope Criterion for
Students are asked to prove that if the slopes of two lines are
Perpendicular Lines - 2
both opposite and reciprocal, then the lines are perpendicular.
Writing Equations for Parallel Students are asked to identify the slope of a line parallel to a
Lines
given line and write an equation for the line given a point.
Writing Equations for
Students are asked to identify the slope of a line perpendicular
Perpendicular Lines
to a given line and write an equation for the line given a point.
Worksheet
Name
Midpoints of the Sides of a
Quadrilateral
Description
The students will construct a quadrilateral on graph paper,
determine the midpoints of each of the four sides, then connect
the midpoints of adjacent sides. The question then is the
following: what are the properties of the resulting quadrilateral?
Students need to justify their conclusions.
Tutorial
Name
Parallel Lines
Perpendicular Lines
Video/Audio/Animation
Description
Parallel lines have the same slope and no points in common.
However, it is not always obvious whether two equations
describe parallel lines or the same line.
Perpendicular lines have slopes which are negative reciprocals
of each other, but why?
Name
Description
This video illustrates how to determine if the graphs of a given
set of equations are parallel.
This video shows how to determine which lines are parallel
from a set of three different equations.
This video describes how to determine the equation of a line
that is perpendicular to another line. All that is given initially
the equation of a line and an ordered pair from the other line.
Parallel Lines
Parallel Lines 2
Perpendicular Lines 2
Assessment
Name
Sample 1 - High School
Geometry State Interim
Assessment
Sample 2 - High School
Geometry State Interim
Assessment
Sample 3 - High School
Geometry State Interim
Assessment
Sample 4 - High School
Geometry State Interim
Assessment
Description
This is a State Interim Assessment for 9th-12th grade.
This is a State Interim Assessment for 9th-12th grade.
This is a State Interim Assessment for 9th-12th grade.
This is a State Interim Assessment for 9th-12th grades.
Student Resources
Name
A Midpoint Miracle
Parallel Lines
Parallel Lines
Parallel Lines 2
Perpendicular Lines
Description
This problem solving task gives students the opportunity to prove a fact
about quadrilaterals: that if we join the midpoints of an arbitrary
quadrilateral to form a new quadrilateral, then the new quadrilateral is a
parallelogram, even if the original quadrilateral was not.
Parallel lines have the same slope and no points in common. However,
it is not always obvious whether two equations describe parallel lines or
the same line.
This video illustrates how to determine if the graphs of a given set of
equations are parallel.
This video shows how to determine which lines are parallel from a set
of three different equations.
Perpendicular lines have slopes which are negative reciprocals of each
other, but why?
This video describes how to determine the equation of a line that is
Perpendicular Lines 2 perpendicular to another line. All that is given initially the equation of a
line and an ordered pair from the other line.
Triangles inscribed in This problem solving task challenges students to use ideas about linear
a circle
functions in order to determine when certain angles are right angles.
Unit Squares and
This problem solving task asks students to find the area of a triangle by
Triangles
using unit squares and line segments.
Parent Resources
Name
Description
This problem solving task gives students the opportunity to prove a fact
about quadrilaterals: that if we join the midpoints of an arbitrary
A Midpoint Miracle
quadrilateral to form a new quadrilateral, then the new quadrilateral is a
parallelogram, even if the original quadrilateral was not.
Triangles inscribed in This problem solving task challenges students to use ideas about linear
a circle
functions in order to determine when certain angles are right angles.
Unit Squares and
This problem solving task asks students to find the area of a triangle by
Triangles
using unit squares and line segments.