Download MAFS.912.G-GPE.2.5 - Prove the slope criteria for parallel and

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
Date Adopted or Revised: 02/14
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.
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
1206310:
1206315:
1206320:
7912065:
1200400:
1207310:
1200410:
1200700:
Course Title
Geometry (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
Geometry Honors (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
Access Geometry (Specifically in versions: 2015 and beyond (current))
Intensive Mathematics (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
Liberal Arts Mathematics (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
Mathematics for College Readiness (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
Related Access Points
Access Point
Access Points Number
MAFS.912.G-GPE.2.AP.5a:
MAFS.912.G-GPE.2.AP.5b:
Access Points 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
Description
Students will be guided through the investigation of y = mx+b. Through this lesson, students will be able to determine
"When will we ever meet?":
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
Forget Waldo - Where is 'the them. The students will repeat this process three times - using different colors for differentiating one line from the next.
orthocenter'?:
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
page 1 of 3 Graphing Equations on the
Cartesian Plane: Slope:
Investigating Lines With Our
Minds!:
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, students learn how slope relates to parallel
and perpendicular 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 the slopes of parallel and perpendicular lines using X-Y
Coordinate Pegboards.
Problem-Solving Task
Name
A Midpoint Miracle:
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.
This problem solving task challenges students to use ideas about linear functions in order to determine when certain
Triangles inscribed in a circle: angles are right angles.
Unit Squares and Triangles:
This problem solving task asks students to find the area of a triangle by using unit squares and line segments.
Formative Assessment
Name
Finding Equations of Parallel
and Perpendicular Lines:
Description
This lesson is intended to help you assess how well students are 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
Parallel Lines - One:
Proving Slope Criterion for
Parallel Lines - Two:
Proving Slope Criterion for
Perpendicular Lines - 1:
Students are asked to prove that two parallel lines have equal slopes.
Proving Slope Criterion for
Perpendicular Lines - 2:
Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are
perpendicular.
Writing Equations for Parallel
Lines:
Students are asked to identify the slope of a line parallel to a given line and write an equation for the line given a
point.
Writing Equations for
Perpendicular Lines:
Students are asked to identify the slope of a line perpendicular to a given line and write an equation for the line given
a point.
Students are asked to prove that two lines with equal slopes are parallel.
Students are asked to prove that the slopes of two perpendicular lines are both opposite and reciprocal.
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:
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:
Perpendicular lines have slopes which are negative reciprocals of each other, but why?
Video/Audio/Animation
Name
Parallel Lines:
Description
This video illustrates how to determine if the graphs of a given set of equations are parallel.
Parallel Lines 2:
This video shows how to determine which lines are parallel from a set of three different equations.
Perpendicular Lines 2:
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.
Assessment
Name
Sample 1 - High School
Geometry State Interim
Assessment:
Sample 2 - 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.
page 2 of 3 Sample 3 - High School
Geometry State Interim
Assessment:
Sample 4 - High School
Geometry State Interim
Assessment:
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:
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:
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.
Parallel Lines:
This video illustrates how to determine if the graphs of a given set of equations are parallel.
Parallel Lines 2:
This video shows how to determine which lines are parallel from a set of three different equations.
Perpendicular Lines:
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 to another line. All that is given initially the
Perpendicular Lines 2: equation of a line and an ordered pair from the other line.
Triangles inscribed in a This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles
are right angles.
circle:
Unit Squares and
Triangles:
This problem solving task asks students to find the area of a triangle by using unit squares and line segments.
Parent Resources
Name
A Midpoint Miracle:
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.
Triangles inscribed in a This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles
are right angles.
circle:
Unit Squares and
Triangles:
This problem solving task asks students to find the area of a triangle by using unit squares and line segments.
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