Download Triangle Proportionality Theorem

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 71668
Triangle Proportionality Theorem
Students are asked to prove that a line parallel to one side of a triangle divides the other two sides of the triangle proportionally.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, triangle proportionality theorem, side splitter theorem, proof
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_TriangleProportionalityTheorem_Worksheet.docx
MFAS_TriangleProportionalityTheorem_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 proof on the Triangle Proportionality Theorem worksheet.
2. The teacher asks follow-up questions, as needed.
Note: Remind the student that a theorem cannot be used in its own proof. If necessary, review the theorems that have been proven prior to the introduction of this
theorem, so the student understands which theorems can be used in this proof.
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:
States the given information but is unable to go further.
page 1 of 4 States the Triangle Proportionality Theorem as a justification of the statement to be proven.
Writes several statements that may or may not be relevant or make sense.
Establishes that
but goes no further.
Questions Eliciting Thinking
What do you know about this figure? What are you being asked to prove?
What was the plan you developed for your proof before starting?
What do you know about parallel lines that could help you? If two parallel lines are intersected by a transversal, what do you know about the angles formed?
Would identifying similar triangles lead you to proportional parts? How?
Instructional Implications
Assist the student in understanding what can be used in a proof of a theorem (i.e., the assumptions presented in the statement of the theorem as well as definitions,
postulates, and other previously established theorems). Emphasize that a theorem cannot be used as a justification in its own proof. Provide feedback to the student
concerning any assumptions made that are not warranted such as assuming that the triangle is isosceles. Assist the student in devising an overall strategy for the proof: (1)
Show the two triangles are similar to conclude that
. (2) Reason from
to
. Then review the theorems, postulates and definitions
needed to complete the proof (e.g., AA Similarity Theorem, Corresponding Angles Theorem, Segment Addition Postulate, Substitution Property of Equality, Reflexive
Property, Subtraction Property of Equality). Guide the student through the statements of the proof and prompt the student to supply the justifications.
Provide proofs of other theorems in which the statements and reasons are given separately and the student must rearrange the steps into a logical order (e.g., a proof of
the Triangle Midsegment Theorem). Consider using the NCTM lesson Pieces of Proof (http://illuminations.nctm.org/Lesson.aspx?id=2561). Allow the student to work with a
partner to complete these exercises.
Encourage the student to begin the proof process by developing an overall strategy. Provide opportunities for the student to determine the flow of a proof. Provide
another statement to be proven and have the student compare strategies with another student and collaborate on completing the proof.
Consider implementing other MFAS proof tasks, Isosceles Triangle Proof (G-CO.3.10), Triangle Sum Proof (G-CO.3.10), or Triangle Midsegment Proof (G-CO.3.10). If needed,
provide the student with the steps of the proof with several of the statements and/or justifications missing.
Moving Forward
Misconception/Error
The student’s proof shows evidence of an overall strategy but fails to establish major conditions leading to the prove statement.
Examples of Student Work at this Level
The student:
Establishes that
and then concludes
Does not understand how to use
.
to conclude that
or reason from
to
.
Questions Eliciting Thinking
If
, what can you say about the sides of the triangles? Are
and
sides of these triangles?
page 2 of 4 How can we use the proportion resulting from the similar triangles,
, to prove
? What is the relationship between AF, FB, and AB?
Instructional Implications
Review the theorems, postulates and definitions needed to complete the proof (e.g., AA Similarity Theorem, Corresponding Angles Theorem, Segment Addition Postulate,
Substitution Property of Equality, the definition of similarity for triangles, Reflexive Property, Subtraction Property of Equality). Review an overall strategy for the proof and
guide the student through the steps of any aspect of his or her proof that was incomplete. Prompt the student to provide justifications for each step.
Encourage the student to begin the process of writing a proof by developing an overall strategy and have the student compare his or her strategies with the strategies of
another student at the same level. For additional practice, provide other theorems to be proven in which the statements and reasons are given separately and the student
must rearrange the steps into a logical order.
Consider implementing other MFAS proof tasks, Isosceles Triangle Proof (G-CO.3.10), Triangle Sum Proof (G-CO.3.10), or Triangle Midsegment Proof (G-CO.3.10).
If necessary, review notation for naming angles (e.g.,
instead of
) when more than one angle share a common vertex. Also, review that when naming similar
triangles, vertices are named in corresponding order.
Almost There
Misconception/Error
The student’s proof shows evidence of an overall strategy, but the student fails to establish a minor condition that is necessary to prove the theorem.
Examples of Student Work at this Level
The student:
Does not clearly indicate the two congruent angle pairs in order to conclude that
by the AA Similarity Theorem.
Does not clearly state that AB = AF + FB and AC = AG + GC and then uses the Segment Addition Postulate to justify a substitution.
Shows that
follows from
.
Questions Eliciting Thinking
Did you justify all of your statements? Is there any additional explanation or reasoning you can provide?
I see you stated these triangles are similar. Can you show me all of the steps needed to use the AA Similarity theorem? Did you include all of them in your proof?
Why are you able to replace AB with AF + FB? Do the statements you included justify your reasoning?
How does
follow from
?
Instructional Implications
Provide the student with feedback on his or her proof. If the student omitted a statement, have the student go through each step of the proof to see if he or she can
find the gap in the logical flow of the proof. Prompt the student to supply justifications or statements that are missing. If needed, review the theorems, postulates and
definitions needed to complete the proof (e.g., AA Similarity Theorem, Corresponding Angles Theorem, Segment Addition Postulate, Substitution Property of Equality,
Corresponding Sides of Similar Triangles are Proportional, Reflexive Property, Subtraction Property of Equality). Encourage the student to correct any misuse of notation,
including misleading abbreviations of justifications. If necessary, review notation for naming angles (e.g.
instead of
, when more than one angle shares a common
vertex). Also, review that when naming similar triangles, vertices are named in corresponding order.
Consider implementing MFAS tasks Isosceles Triangle Proof (G-CO.3.10), Triangle Sum Proof (G-CO.3.10), and Triangle Midsegment Proof (G-CO.3.10).
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 reasons from the given that either two pairs of corresponding angles formed by the parallel lines are congruent or that one pair of corresponding angles is
congruent and
because of the Reflexive Property. The student then concludes that
because of the AA Similarity Theorem. The student
writes that because corresponding sides of similar triangles are proportional that
. The student explains that AB = AF + FB and AC = AG + GC because of the
Segment Addition Postulate. By using the Substitution Property of Equality the student writes that
. The student then reasons from this
page 3 of 4 proportion to
. For example, the student shows that:
AF(AG + GC) = AG(AF + FB)
and cites appropriate
justifications.
Questions Eliciting Thinking
Could you have reasoned from
to
in another way? Explain why or why not?
What other proportions are true in the diagram that is given?
Instructional Implications
Encourage the student to use the Triangle Proportionality Theorem to prove the corollary that states that if three or more parallel lines are intersected by two transversals,
the transversals are divided into proportional parts by the parallel lines.
Consider using MFAS task Converse of the Triangle Proportionality Theorem (G-SRT.2.4) if not previously used.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Triangle Proportionality Theorem 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
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-SRT.2.4:
Description
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
page 4 of 4