Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 57196
Triangle Midsegment Proof
Students are asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its
length.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, triangle midsegment, proof, parallel
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_TriangleMidsegmentproof_worksheet.docx
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 Triangle Midsegment Proof 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 attempts to calculate the coordinates of the midpoints D and E, but is unable to do so correctly and goes no further.
page 1 of 4 Questions Eliciting Thinking
It looks like you were trying to calculate the coordinates of midpoints D and E. Why did you do that? What should you do next?
Can you tell me, in general, how you can prove segments or lines are parallel?
What would you need to do to show that DE =
AC?
Instructional Implications
Provide the student with frequent opportunities to make deductions using a variety of previously encountered definitions and established theorems. For example, provide
diagrams as appropriate and ask the student what can be concluded as a consequence of:
Point M is the midpoint of
.
< A and < B are supplementary and m < A = d.
and PQ = m units.
is the bisector of < E.
Provide the student with the statements of a proof and ask the student to supply the justifications. Then have the student analyze and describe the overall strategy used
in the proof.
Review the formulas needed to complete this proof (i.e., the midpoint formula, the distance formula, and the slope formula).
Moving Forward
Misconception/Error
The student’s proofs shows no evidence of an overall strategy but upon questioning the student can explain the general steps of the proof.
Examples of Student Work at this Level
The student makes little or no progress on the proof. The student’s work indicates the use of either the slope formula or the distance formula, but complete work and
explanations are missing. Upon questioning, the student explains that (1) the segments can be shown to be parallel by showing their slopes are the same and (2) DE and
AC must be determined to show that DE =
AC.
Questions Eliciting Thinking
What work should you show to prove
What work is needed to prove DE =
parallel to
?
AC?
Instructional Implications
Review the formulas needed to complete this proof (i.e., the midpoint formula, the distance formula, and the slope formula). Assist the student in showing that
parallel to
is
. Model presenting steps in a logical order, labeling all work, presenting computations clearly and completely, explicitly stating the conclusion, and removing any
extraneous work left on the paper. Ask the student to provide justifications for each step. Have the student show that DE =
AC. Provide feedback as needed.
Consider using the NCTM lesson Pieces of Proof (http://illuminations.nctm.org/Lesson.aspx?id=2561) in which the statements and reasons of a proof are given separately
and the student must rearrange the steps in a logical order. Encourage the student to use multiple proof formats including flow diagrams, two-column, and paragraph
proofs. 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 another statement to be proven and have the student compare strategies
page 2 of 4 with another student and to collaborate on completing the proof.
Almost There
Misconception/Error
The student exhibits all necessary components of the proof but does not write the proof in a coherent and complete manner.
Examples of Student Work at this Level
The student correctly calculates:
The coordinates of points D and E
The slopes of
and
, and
The lengths, DE and AC,
but fails to label the coordinates and calculated quantities, presents work haphazardly on his/her paper, and/or fails to draw any explicit conclusions.
Questions Eliciting Thinking
Do you think that someone reading your proof could easily understand your work and follow your proof?
What were you calculating here? Can you label this work so that a reader could understand what you were doing?
Does this order make sense to you? Can you calculate the slopes before you know the coordinates of the midpoints?
Are all of your steps justified? Is there any additional explanation or reasoning you can provide?
Instructional Implications
Discuss with the student how to write a clear and convincing proof. Show the student a model coordinate geometry proof of another statement and point out all of the
features that make it clear and convincing (e.g., steps are presented in a logical order, all work is labeled, computations are clearly presented, conclusions are explicitly
stated, and no extraneous work is left on the paper). Have the student rewrite his or her proof so that it is clear and convincing. Then have the student trade papers with
another Almost There student in order to analyze each other’s work and provide feedback.
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 finds and labels the coordinates of point D as (b, c) and point E as (a + b, c). The student uses the slope formula to calculate the slopes of
order to show that they are parallel and uses the distance formula to calculate both DE and AC in order to show DE =
and
in
AC. Appropriate justification is given throughout
the proof.
Questions Eliciting Thinking
There is just one calculation on your paper that you neglected to label? Can you label that so a reader will know exactly what you were calculating?
How could you use similar triangles to show
and DE =
AC?
Instructional Implications
page 3 of 4 Challenge the student with statements requiring more complex proofs (e.g. given a diagram that includes overlapping triangles, ask the student to prove a statement that
requires first proving one pair of triangles congruent in order to name a pair of corresponding parts congruent needed to show a second pair of triangles congruent).
Encourage the student to assist other students in developing and writing proofs.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Triangle Midsegment Proof 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-CO.3.10:
Description
Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of
interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent;
the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
page 4 of 4