Download Proving Slope Criterion for Parallel Lines - Two

Proving Slope Criterion for Parallel Lines Two
Resource ID#: 72008
Primary Type: Formative Assessment
This document was generated on CPALMS - www.cpalms.org
Students are asked to prove that two lines with equal slopes are parallel.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, parallel, slope, proof, similar triangles, slope triangles
Instructional Component Type(s): Formative Assessment
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ProvingSlopeCriterionForParallelLines-2_Worksheet.docx
MFAS_ProvingSlopeCriterionForParallelLines-2_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problem on the Proving Slope Criterion for
Parallel Lines - Two worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student’s proof shows no evidence of an overall strategy or logical flow.
Examples of Student Work at this Level
The student:
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
States the given information but is unable to go any further.
Uses criteria that have yet to be proven.

Makes some observations about lines with equal slopes without regard to the statement to be proven.
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States that the lines are parallel without any reasoning.
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Draws two lines with equal slopes and proves that those two specific lines are parallel.
Questions Eliciting Thinking
What information were you given? Did you sketch lines a and b?
What can you assume is true about lines a and b? What are you being asked to prove?
Did you think of a plan for your proof before you started?
I see that you found the slope of each of your lines and said they were equal. Did you prove only that your two p
Instructional Implications
Describe an overall strategy for the proof (e.g., draw a diagram that includes "slope triangles," use the fact that t
the consequences of similarity to show that a pair of angles are congruent that will lead to the conclusion that the
statements of the proof and ask the student to supply the justifications.
Remind the student that the statement to be proven is general and applies to any pair of non-vertical parallel line
variables instead of specific values.
Provide the student with a diagram that includes a pair of lines with equal slopes along with appropriately drawn
convincing mathematical argument that shows the two lines are parallel.
Moving Forward
Misconception/Error
The student’s proof shows evidence of an overall strategy but fails to establish major conditions leading to the p
Examples of Student Work at this Level
The student sketches lines a and b and constructs two transversals. The student attempts to show that lengths of



States the lengths of the sides are proportional without proof.
States the triangles are similar without proof.
Proves the triangles are similar but then assumes the lines are parallel.
Questions Eliciting Thinking
What do you know about the slopes of these lines? Can you write that in a proportion?
What do you know about similar triangles? How can you prove two triangles are similar?
Can you mark which angles are congruent? What type of angles are these?
What must be true if the alternate interior angles are congruent?
Instructional Implications
Review the overall strategy used in the student’s proof and provide feedback concerning any aspect of the proof
Guide the student to observe that the slopes of lines a and b can be written as a ratio of corresponding sides of th
corresponding sides are proportional. Remind the student that once two triangles are proven similar, then the cor
Ask the student to use the slope triangles drawn in the diagram to write expressions for the slope of each line. G
proportionality of the sides. Make sure the student understands that this needs to be explicitly stated in the proof
Address any misuses of notation (e.g., confusing measures of angles with their names, naming an angle with one
Provide the student with a diagram that includes a pair of lines with equal slopes along with appropriately drawn
convincing mathematical argument that shows the two lines are parallel.
Almost There
Misconception/Error
The student provides a correct response but with insufficient reasoning or imprecise language.
Examples of Student Work at this Level
The student sketches lines a and b and draws two transversals to form slope triangles. The student writes a propo
that that the corresponding angles are congruent, but:



Misuses notation.
Neglects to state that VWX and XYZ are corresponding angles of similar triangles and are, therefore,
Provides an incorrect justification (e.g., justifies the congruence of two vertical angles by citing the defin
Questions Eliciting Thinking
There is a small error in your proof. Can you find it?
Why are the lines parallel? Did you state that in your proof?
Instructional Implications
Provide feedback to the student concerning any errors made and allow the student to revise his or her proof.
Address any misuses of notation (e.g., confusing measures of angles with their names, naming an angle with one
Challenge the student with the MFAS tasks Proving Slope Criterion for Parallel Lines - One (G-GPE.2.5), Prov
Criterion for Perpendicular Lines - Two (G-GPE.2.5).
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student provides a complete proof with justification such as:
Given lines a and b, draw a vertical and a horizontal transversal that intersect each other between the lines as sho
can be given as
. Since the slopes are equal (by assumption), then
=
. Also VXW
YXZ by the Ve
Theorem. Since VWX and XYZ are corresponding angles of similar triangles, VWX
XYZ. Consequently,
Theorem).
Questions Eliciting Thinking
Is there another method you could have used to prove that lines with the same slope are parallel?
Were there any statements in your proof that you did not really need?
Instructional Implications
Ask the student to devise a coordinate geometry proof to show that lines with the same slope are parallel.
Challenge the student with MFAS tasks Proving Slope Criterion for Parallel Line - One (G-GPE.2.5), Proving S
Criterion for Perpendicular Lines - Two (G-GPE.2.5).
ACCOMMODATIONS & RECOMMENDATIONS

Special Materials Needed:
o
Proving Slope Criterion for Parallel Lines - Two worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-GPE.2.5:
Description
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).
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.