Download Equidistant Points

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 60313
Equidistant Points
Students are asked to prove that a point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, perpendicular bisector, equidistant
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_Equidistant Points_Worksheet.docx
MFAS_Equidistant Points_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 Equidistant Points 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:
States the given and one or two more statements that fail to establish that
or that
.
page 1 of 4 Offers an explanation that is unclear or does not make mathematical sense.
Restates the statement to be proved.
Questions Eliciting Thinking
What is it that you are given and what are you trying to prove?
Did you think through a plan for your proof before you started? Did you consider what you already know that might help you to prove that point P is equidistant from points
A and B?
You referred to an angle as
AP. What angle did you mean? How are angles typically named?
What do you know as a consequence of the fact that
is the perpendicular bisector of
?
Instructional Implications
Provide the student with the statements of a proof of this theorem and ask the student to supply the justifications. Then have the student analyze and describe the overall
strategy used in the proof. If needed, review the ways to prove two triangles are congruent (SSS, SAS, ASA, AAS, and HL) and what must be established in order to
conclude two triangles are congruent when using each method. Remind the student that once two triangles are proven congruent, all remaining pairs of corresponding
parts can be concluded to be congruent.
Emphasize that a theorem cannot be used as a justification in its own proof. Encourage the student to first identify the assumptions given in the statement to be proved
along with other definitions, postulates, and theorems already established.
Provide the student with 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
and
.
are supplementary and
= d.
and PQ = m units.
is the bisector of
.
Moving Forward
Misconception/Error
The student’s proof shows evidence of an overall strategy but the student fails to establish major conditions leading to the prove statement.
Examples of Student Work at this Level
The student:
Adds auxiliary segments (
and
) and then marks congruent segments and right angles in the diagram, and writes only, “By SAS.” No proof is provided. Adds auxiliary segments (
and
), states
with no supporting statements and concludes that
.
Questions Eliciting Thinking
What do you know as a consequence of the fact that
is the perpendicular bisector of
?
page 2 of 4 How can you show
?
Instructional Implications
Ask the student to indicate what it means for
to be the perpendicular bisector of
. Prompt the student to consider each word, perpendicular and bisector, separately
and to mark the diagram accordingly. Guide the student to write a statement that corresponds to each of the marks made on the diagram and to justify them. Provide
additional feedback to the student regarding any other missing statements or justifications. Review the triangle congruence theorems and provide more opportunities and
experiences with proving triangles congruent.
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
with another student and to collaborate on completing the proof.
Almost There
Misconception/Error
The student’s proof shows evidence of an overall strategy, but the student fails to establish a condition that is necessary for a later statement in the proof.
Examples of Student Work at this Level
The student fails to:
Justify adding auxiliary segments (
Establish the congruence of
and
and
Explicitly state that
State that
) to the diagram.
.
but provides all supporting statements.
after showing that
.
Questions Eliciting Thinking
How did you know you could add these segments to your diagram? How do you know these segments are unique?
How do you know
?
Were you trying to prove
? Should this be stated in your proof?
Why does the congruence of
and
imply congruence of
and
?
Instructional Implications
Review how to address and justify adding an auxiliary line or segment to a diagram.
Using a colored pencil or highlighter, encourage the student to mark the statements which support the congruence theorem chosen. Remind the student that each letter
of the theorem name represents a pair of parts that must be shown to be congruent. For example, if using SAS to prove the triangles congruent, the proof must include
showing two pairs of corresponding sides and their included angle are congruent (and a reason or justification must be provided for each).
When proving statements and theorems, ask the student to review his or her proof to be sure that the assumptions implicit in the statement were used as justifications and
that the proof concludes with the statement to be proven.
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 devises a complete and correct proof in which he or she (1) Adds auxiliary segments (
is the perpendicular bisector of
to show
by SAS, and (3) Concludes that
and
) so that two triangles are formed, (2) Uses the fact that
by definition of congruent triangles.
Questions Eliciting Thinking
What if
was not perpendicular to
? What can you say about the distance from point P to points A and B?
Instructional Implications
Pose the following problem variation for the student: Suppose point P is closer to A than it is to B. Determine and justify which is greater, AP or BP.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Equidistant Points worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
page 3 of 4 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.3.9:
Description
Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include:
vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant
from the segment’s endpoints.
page 4 of 4