Triangle Inequality Theorem: Activities and Assessment Methods

EDG544 Assessment Project (Template)
Name: Marianne McFadden
Subject Area: Mathematics – Geometry (non-honors)
Grade Level: Grades 8 – 11
AACCHHIIEEVVEEMMEENNTT TTAARRGGEETT::
APPLYING THE TRIANGLE INEQUALITY THEOREM AND THE PYTHAGOREAN THEOREM (WITH ITS CONVERSES)
EDUCATOR LANGUAGE (PA Academic Standards):
M11.C.1.2.1 Identify and prove the properties of triangles involving opposite sides and angles.
M11.C.1.4.1 Identify and prove properties of right triangles using the Pythagorean Theorem; use
converse forms of the Pythagorean Theorem to classify (by angles) types of triangles specified.
STUDENT-FRIENDLY LANGUAGE:
I can look at any triangle with the side lengths labeled and list the angles from smallest to largest. I
can look at any triangle with the angle measures labeled and list the sides from shortest to longest.
Also, I can decide if three measures that I am given will be lengths that can form a triangle.
The angle-side relationships we studied and the ‘triangle inequality rules’ show me how to figure out
these kinds of problems.
I can take three side lengths that I am given and figure out what kind of triangle will be formed, if I
connect the lengths tip to tip. If I use the Pythagorean Theorem to show I get a true statement, then
I form a right triangle. If I use one of the converses of the Theorem to arrive at a true statement,
then either an acute or obtuse triangle is formed.
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►► TTRRIIAANNGGLLEE IINNEEQQUUAALLIITTYY TTHHEEOORREEMM ◄◄
◄◄ PPYYTTHHAAGGOORREEAANN TTHHEEOORREEMM ◄◄
((AANNDD CCOONNVVEERRSSEE FFOORRMMSS))
KNOWLEDGE
▪ identify parts (sides and angles) of a triangle
▪ classify triangles according to sides (scalene, isosceles, equilateral)
and angles (acute, right, obtuse, equiangular)
▪ define perimeter; state perimeter rule for triangles
▪ identify side opposite angle specified and angle opposite side
specified (side/angle relationships)
▪ state Triangle Inequality Theorem
▪ identify parts of a right triangle: hypotenuse, legs, right angle,
acute angles
▪ state Pythagorean Theorem and its converse
▪ state the two converse forms (> or <), then restate the
appropriate conclusion for each
▪ list example integral (whole number) values for which the
Pythagorean Theorem and its converse forms hold true
REASONING
▪ interpret angle measures of a triangle in order to determine a
shortest-to-longest order for the side measures; interpret side
measures to determine a smallest-to-largest order for the angle
measures
▪ explain the difference between a perfect square and a non-
perfect square
▪ explain how radical values are squared; find the square of a
radical value
PERFORMANCE SKILLS
▪ explain the difference between three values that could represent the
sides of a triangle and three values that cannot represent the side
lengths of a triangle (by using the triangle inequality theorem)
▪ given a specified perimeter, construct a chart illustrating all possible
combinations of integral side lengths that could possibly represent the
sides of a triangle with perimeter specified
▪ compute the squares of radical values
▪ given three integral values, compute their squares and arrange
results in descending order in setting values up for converse forms
of Pythagorean Theorem
▪ given three values (including one or more that contain radicals),
compute their squares and arrange results in descending order

PRODUCT LEARNING
▪ analyze chart values in demonstrating, by applying the Triangle
Inequality Theorem, which set(s) of values determine the lengths of a
triangle; provide counterexamples for the theorem from the chart
values; provide computational support for each conclusion stated
▪ given TWO integral values, determine a range of possible values for
a third value in order for a triangle to be formed; provide computational
support for answers stated
▪ apply Triangle Inequality Theorem in solving word problems that
involve missing side lengths
▪ construct triangles with given specifications that require possible
missing lengths be determined before constructing the figures
▪ evaluate chart data that determine a triangle --determine what
type of triangle is formed from each set of values that DO
determine a triangle; provide computational support for each
conclusion stated
▪ given TWO integral values, test a third value that would
determine the side lengths for an acute triangle and test another
third value that would determine the side lengths of an obtuse
triangle; provide computational support (if one or both conditions
are not possible, determine and explain why)
▪ apply the converse forms of the Pythagorean Theorem in solving
word problems that involve missing side lengths
▪ determine properties (type of Δ, by sides and angles) of the
triangles constructed (see ‘construct’ in left column, last skill listed)
DISPOSITIONS
▪ respond to a survey, at the end of the unit, that determines student’s
comfort and proficiency in applying the Triangle Inequality Theorem
and how it was utilized in the performance rubric activity assessment.
▪ respond to a survey, at the end of the unit, that determines
student’s comfort and proficiency in applying the Pythagorean
Theorem (and converses) and how they were utilized in the
constructed response assessment.
SSTTUUDDEENNTTSS WWHHOO AARREE MMAAKKIINNGG AAPPPPRROOXXIIMMAATTIIOONNSS TTOOWWAARRDD TTHHEE TTAARRGGEETT
SSHHOOUULLDD PPRROOBBAABBLLYY BBEE AABBLLEE TTOO::
 Summary and differences between ‘successful’ chart and ‘approximations’ chart:
Those students making approximations toward the target will NOT be required to:
 work with non-whole number values, work with problems having missing values, solve more difficult
applications problems by themselves
More emphasis for these students (mostly learning support) will be placed on:
 utilizing PSSA formula sheets regularly, expressing concepts in the student’s own words, giving a
verbal explanation of ideas learned, working cooperatively in teams or small groups, journaling to
self-assess and/or express concerns about concepts that need more practice
⇛⇛ TTRRIIAANNGGLLEE IINNEEQQUUAALLIITTYY TTHHEEOORREEMM ⇚⇚
⇛⇛ PPYYTTHHAAGGOORREEAANN TTHHEEOORREEMM ⇚⇚
((AANNDD CCOONNVVEERRSSEE FFOORRMMSS))
KNOWLEDGE
▪ identify parts (sides and angles) of a triangle
▪ classify triangles according to sides (scalene, isosceles, equilateral)
and angles (acute, right, obtuse, equiangular)
▪ define perimeter & state perimeter rule for triangles, using PSSA
rule sheet available in the classroom
▪ identify side opposite angle specified and angle opposite side
specified (side/angle relationships)
▪ state Triangle Inequality Theorem in student’s own words
▪ identify parts of a right triangle (hypotenuse, legs, right angle,
acute angles)
▪ state Pythagorean Theorem and its converse in student’s own
words
▪ state the two converse forms (> or <), then describe the
appropriate conclusion for each, in student’s own words
▪ list TWO example sets of integral (whole number) values for
which the Pythagorean Theorem and its converse forms hold true
(values are memorized as a class exercise)
REASONING
▪ interpret given angle measures of a triangle in order to determine a
shortest-to-longest order for the side measures; interpret given side
measures to determine a smallest-to-largest order for the angle
measures – state relationship between side lengths and angle
measures in student’s own words
▪ explain the difference between values given that are perfect
squares and those given that are non-perfect squares, in
student’s own words
PERFORMANCE SKILLS
▪ explain the difference between three values that represent the sides
of a triangle and three values that do not represent the side lengths of
a triangle (by using the triangle inequality theorem) when given one set
▪ given three whole number values, compute their squares and
arrange results in descending order in setting values up for
converse forms of Pythagorean Theorem

of values for each category
PRODUCT LEARNING
▪ analyze chart values in demonstrating, by applying the Triangle
Inequality Theorem, which set(s) of values determine the lengths of a
triangle; provide computational support for each conclusion stated
▪ given TWO integral values, determine two possible values for a third
side length in order for a triangle to be formed; provide computational
support for answers stated, then (see column to the right ⇛)
▪ evaluate given chart data that form a triangle by determining
what type of triangle is formed from each set of values that DO
form a triangle; provide computational support for each conclusion
stated
▪ apply the converse forms of the Pythagorean Theorem in solving
word problems
▪ (⇛ from column to the left): determine properties (type of Δ, by
sides and angles) of the triangles formed and verbally describe
reasoning (using converse forms of Pythagorean Theorem)
DISPOSITIONS
▪ self-assess student progress at the end of this unit by completing a
journal entry and then discussing the entry when conferencing with the
teacher (in order to determine student’s comfort and proficiency in
applying the Triangle Inequality Theorem and how it was utilized in the
performance rubric activity assessment).
▪ self-assess student progress at the end of this unit by completing
a journal entry and then discussing the entry when conferencing
with the teacher (in order to determine student’s comfort and
proficiency in applying the Pythagorean Theorem and its converses
and how they were utilized in the constructed response
assessment).
Current Assessments for this Achievement Target:
Assessment*
Type(s) of
Thinking Assessed
Method(s) of
Assessment
(Ch 5-9)
Use of the Data Collected
Pre-Assessment
(terminology
review)
 Knowledge
(#1-19, #22-30)
 Reasoning (#20-21)
selected response (fill in
the blank; multiple
choice)
A guided practice/review of essential
terminology in order to complete unit on
theorems and converses successfully
USE OF DATA/RESULTS:
- results determine whether student has
mastered basic vocabulary and
angle/side relationships in order to begin
study of Triangle Inequality properties
- poor scores would indicate that review
is needed before moving on
Assessment #1
(performance
rubric)
 Performance Skills
(#1, all but last column)
 Product Learning
(#1 – last column,
#2 – 4)
selected response,
extended written
response, performance
assessment (open-
ended; fill in chart of
investigated values)
An investigation of two theorems and
converses using manipulatives
- correctness and completeness of chart
of values determines whether the hands-
on approach (to the theorems
investigated) helps student to better
visualize how a triangle is formed (or
cannot be formed)
- if several students show incomplete
charts, a teacher demo should help
Assessment #2
(selected
response)
 Reasoning (#5, 6, 9, 10,
14, 16 – 18)
selected response
(multiple choice; true
and false)
Questions relating to using
theorems/converse forms without
manipulatives

(#1, 11, 12)
(#2, 3, 4, 7, 8, 13, 15)
- scores of 60% or above indicate that
student is able to use properties in the
theorems studied
- several low scores would indicate that
teacher should model problems so
students can re-do them
Assessment #3
(constructed
response)
 Reasoning (#3, 9)
(#1, 2, 4 – 8)
extended written
response (open-ended
responses)
Difficult, multi-step questions relating to
using theorems/converse forms to
construct figures that fit specific
measures (without manipulatives)
- used as an ‘extra’ – success with these
indicate that student is prepared for SAT-
type questions as well as other post-
secondary entrance test type problems
- class discussion of problems should
occur after all students attempt them
* Attach copies of current assessments
Learning experiences provided to achieve the target:
Learning Experience How I Assess
What I Learn
from the Data
Revisions Needed*
PRE-ASSESSMENT
students complete pre-
assessment as a classwork and/or
homework assignment as a
review exercise
- grade pre-assessment
worksheet; take note of
how long students spend on
worksheet as indication of
how much remembered
- if worksheet is done
with ease and students
volunteer to answer
random questions
quickly, then class is
ready to move to new
theorems; if not, some
review is needed
- allow students to work alone for
most of the class, then collaborate
and compare responses for a few
minutes at the end of class before
assigning the remainder of the
exercises for homework
Short video (visual explanation)
students view short video,
depicting how lengths of sides of
a triangle are related (see
youtube website below, labeled
as #1)
- ask students to describe
what is being shown in the
video – why the sticks
forming the triangle move
and line up in
demonstrating the property
shown
- how the students
interpret and describe the
visual depiction of the
theorem being explained
- how students express
mathematical concepts
on their own terms
- allow students to briefly discuss
the visual with a partner and
encourage them to produce their
own labeled drawing, similar to
what the video shows, then ask
for volunteers to describe their
own drawing, with emphasis on
how side lengths are related
ASSESSMENT #1 (RUBRIC ACTIVITY)
students demonstrate properties
from video by using
manipulatives in constructing
triangles
- use the revised rubric
provided to determine
students’ correctness and
completeness, starting with
the chart in the activity and
continuing with the
- how the students think
mathematically
- how small groups work
together to arrive at an
agreed upon solution
- if the exercises are too
-last three sets of side lengths
should be left blank for students
to generate their own values
(teacher can encourage discovery
by studying patterns)

computational support and
related exercises required
difficult (by listening to
group conversation)
CLASS DISCUSSION
students respond to teacher’s
inquiry about activity, emphasis
on chart entries and
computational support, including
related exercises
- question/answer method
in encouraging students to
make observations about
the rubric activity
- have a rep from each
group demonstrate several
chart examples using the
manipulatives
- from careful
questioning, teacher can
determine if concepts are
understood
- questions can delve
deeper into
understanding and
promote more
challenging problem
solving
- after successful class discussion,
students should view Khan
Academy (see website #3 below)
and complete corresponding quiz
in Khan until five responses in a
row are correct (Khan deems as
mastery)
MODEL PROBLEMS
similar to assessment #2 & #3
questions, teacher shows steps
involved when solving problems
related to the properties studied,
BUT requiring deeper
understanding and ingenuity in
applying properties studied
- students complete
assessment #2 (selected
response) and assessment #3
(constructed response)
- assessment #3 completed as a
group effort; one student from
each group chosen to present a
particular problem (with
explanation support from
group members)
- assessment #2is graded
as a regular test,
indicating students’
mastery of the unit
- assessment #3 used to
assess student’s ability to
work cooperatively in
arriving at an agreed
upon explanation of
difficult applications
- after successful completion of
last two assessments, students
should complete two journal
entries: one requiring their
explanation of video portions,
and the other requiring their
response to related problems;
BOTH entries should be an
integral part of their discussion
during their conference with the
teacher (assessments #4 & #5)
WEBSITES – videos described above:
1) http://www.youtube.com/watch?v=MpSI8g2fOH0&feature=player_detailpage
2) http://www.youtube.com/watch?v=J5IP-OPG8Ck
3) https://www.khanacademy.org/math/geometry/basic-
geometry/triangle_inequality_theorem/v/triangle-inqequality-theorem
* Develop samples (assessments listed in ‘revisions needed’ are assessments #4 & #5, to be
submitted after submitting current assessments)
Analysis of Current Approach to Assessment:
Currently, my students take very traditional-type assessments – tests and quizzes that are short
answer, multiple choice, true/false, and constructed response in nature. Those students who follow
modeled examples and problems in class, complete and check homework problems, and volunteer
requested solutions are those who have a very good chance of performing well on the traditional
assessments taken in class. Question/answer on-the-spot assessing occurs daily and throughout all
lessons presenting new concepts; students’ responses to such questions allow me to determine their
level of understanding and whether I need to generate more examples to enhance understanding or
move on to more difficult applications of the concepts presented.
The only differing type of assessment that has been utilized for this unit is the Performance Rubric,
and this has been used successfully many times in average achieving Geometry classes. In realizing that
students can relate to the hands-on approach (using manipulatives) and in doing so increase their level of
understanding, this topic lends itself perfectly to this rubric activity. As a result of this activity, I have

witnessed students grow mathematically in that their increased understanding motivated them to
attempt related problems, including the very challenging ones.
Since today’s students are used to technology use in their classes, it makes sense to attempt to use
more technology routinely, and incorporating assessments #4 and #5 encourages the use of videos and
also calls for students to write as they think mathematically. The variety that is brought to the classroom
through technology allows the educator to tap into multiple methods of instruction, so learning is
enhanced even more, especially for the non-traditional learner.
In-depth Analysis of One Assessment for the Target (including table of item specifications):
NEW ASSESSMENTS BEING IMPLEMENTED:
Assessment*
Type(s) of
Thinking Assessed
Method(s) of
Assessment
(Ch 5-9)
Use of the Data Collected
Assessment #4
(constructed
responses – journal
entry)
 Reasoning (A, B, C, D1,
D2)
extended written
response, personal
communication (survey;
essay response;
conference with
teacher)
Challenging real-life applications of the
properties of triangles studied
- results indicate student’s ability to
transfer triangle properties to true-to-life
situations
- students to work in teams; then
respond with a ‘team answer’ for each,
while producing a written response in
their Math journal for the exercise
- one student from each team required
to present the problem to the class;
emphasis on clarity and completeness of
explanation
Assessment #5
(extended written
response – open
ended response
during conference
with teacher)
(#1, 2, 3)
 Dispositions (#1, 2, 3)
extended written
response, personal
communication
(conference with
teacher)
summary of properties of triangles
studied, expressed in student’s own
words
- student’s verbal explanation, with
teacher’s prompts during conference
indicates depth of knowledge and
understanding of the relationships
learned in the unit
- if student indicates lack of
understanding, additional video lessons
and/or practice problems should help
The assessments described above (not yet implemented) clearly show a shift from the
traditional paper test and quiz assessment style. They encourage collaboration,
communication, and cooperation as students work together to arrive at agreed upon solutions
and methods of effectively solving difficult problems. Lastly, teachers assess students’
understanding individually as they participate in personal conferences with their teacher.

Development and Use of a Student-Friendly Rubric:
Current rubric and revised, single-point rubric are presented as a part of assessment #1’s evaluation
process (attached). The rubric emphasizes completeness and correctness in following all aspects of the
activity so that the student can understand and apply the theorems easily and then move on to attempt
more difficult applications with success.
Issues Related to Communication of Student Achievement:
Difficulties with these assessment types, both current and new:
 most classes contain a wide range of student abilities, so some students will struggle while others are
ready to move on quickly; this frustrates slow-learners, even though the journaling and conferencing are
designed to help them both self-assess and communicate their understanding in their own words
 most students will enjoy the performance rubric (assessment #1), even the slow-learners, but some may
tire of the repetition in the chart investigation and err in filling in values
 even though the theorems are not too difficult to understand, some of the applications in assessments #2
and #3 require higher level thinking – group work will allow faster learners to help slower learners, but
sometimes the faster learners complete most of the work, so the slower learner may not benefit as much
as he/she should from the exercise
 some students may be intimidated from the personal conference with the teacher, others may be
intimidated by presenting mathematical problems to the class as a whole (very different from other
disciplines)
These difficulties may cloud the teacher’s ability in accurately assessing a student’s true progress.
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PRE-ASSESSMENT, and ASSESSMENTS #1, #2, and #3 follow.

Pre-assessment: Review of prerequisite terminology (selected and constructed responses)
Triangles – Review of Terminology: “opposite” and “included”
When finding a side opposite of an angle, look for the side that is NOT one of
the rays that makes up the angle, as shown in the triangle on the left.
State the sides opposite the indicated angles:
1. _______ ∠B 3. _______ ∠F
2. _______ ∠C 4. _______ ∠D
When finding an angle opposite of a side, look for the angle that does
not have the side indicated as one of its rays. In the figure to the left,
∠C is opposite of AB.
Using the figure to the left, state the angle that is opposite the
indicated sides:
5. ________ CA 6. ________ BC
Using the figure to the left, state the following:
In ∆CDB, find the angle opposite of:
7. ________ DB 8. ________ CB
In ∆ABD, find the side opposite of:
9. ________ ∠ADB 10. ________ ∠ABD
11. ________ The side opposite of ∠C in ∆CDB and ∠A in ∆DAB is ?

When finding a side included between two angles, look
for the side (ray) that the angles share in common. In
∆ABC, AC is the side included between ∠A and ∠C.
Using ∆DEF above, find the side included between:
12. ________ ∠E and ∠F 13. ________ ∠D and ∠E
When finding the angle included between two sides, look for the angle that is formed by joining the two
sides named. In ∆DEF above, ∠E is the angle included between DE and EF because these sides form ∠E.
Using ∆ABC above, find the angle include between:
14. ________ AC and AB 15. ________ BC and AC
Using the figure to the left, state the following:
In ∆DAB, find the side included between:
16. ________ ∠DAB and ∠ABD 17. ________ ∠ADB and ∠BAD
In ∆CDB, find the angle included between:
18. ________ DC and CB 19. ________ DB and CD
▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪
Triangles – Review of Ordering Sides and Angles (use of “opposite” terminology)
Review of theorems 6.2 and 6.3:
6.2 → If one side of a triangle is longer than a second side, then the angle opposite the first side is
larger than the angle opposite the second side. 6.3 → If one angle of a triangle is larger than a second
angle, then the side opposite the first angle is longer than the side opposite the second angle.
20. Consider ∆XYZ, where XY = 6, YZ = 7, and XZ = 8. List the angles in order from smallest to largest.
(A labeled sketch will help you). ∠_____ (smallest) ∠_____ ∠____ (largest)
21. Consider ∆PQR, where m∠P = 50, m∠Q = 100. Find m∠R. List the sides in order from shortest to
longest. (A labeled sketch will help you). m∠R = _____; ______(shortest) ______ ______ (longest)

Triangles – Review of Classification (by Angles and Sides)
FIGURE 1 FIGURE 2
FIGURE 3 FIGURE 4 FIGURE 5
For the figures shown, choose the best description of each from the choices given:
22. __Figure 1: 23. __Figure 2: 24. __Figure 3: 25. __Figure 4:
A. acute ∆ A. isosceles ∆ A. acute ∆ A. acute ∆
FIGURE 6 B. obtuse ∆ B. scalene ∆ B. obtuse ∆
B. obtuse ∆
C. right ∆ C. equilateral ∆ C. right ∆ C. right ∆
26. Figure 5: A. isosceles ∆ B. scalene ∆ C. equilateral ∆
27. Figure 6: A. isosceles ∆ B. scalene ∆ C. equilateral and equiangular ∆
Select TWO classifications (one for angles/one for sides) for the following ∆s, given that the aannggllee mmeeaassuurreess are:
28. ___ ___ 90°, 45°, 45° (angles) A. acute B. right C. obtuse D. equiangular
29. ___ ___ 60°, 70°, 50° (sides) E. scalene F. isosceles G. equilateral
30. ___ ___ 100°, 40°, 40°

Assessment #1 – Tool: Performance Rubric
Content/Curriculum Unit Lesson: Geometry/Triangle Inequality Theorem; Pythagorean
Theorem and its Converse Forms
Activity Description: Given a standard 12” pipe cleaner, students will form, by bending the
cleaner at two locations and connecting the ends, all possible triangles with side lengths of
integral (whole number) values. Then students will record observations made in a given chart.
For each set of values considered, students will apply the Triangle Inequality Theorem to
support their conclusion as to whether a triangle can be formed using the given combination.
For those values that DO determine a triangle, students will further apply the Pythagorean
Theorem and its converse forms to determine what type of triangle the values describe. For
those values that do NOT determine a triangle, students will apply the Triangle Inequality
Theorem in supplying computational support as they submit these cases as counterexamples.
Activity Materials: Two twelve inch pipe cleaners, permanent marker to mark spacing of one-
inch sections, worksheet to complete observations and draw conclusions
Evaluation: A rubric will be used in evaluating students’ observations and work. The following
criteria will be considered: use of Mathematical concepts and reasoning in testing each case
specified, use of manipulatives (pipe cleaners), explanation and checking of each case studied,
completion of problems (related exercises), use of Mathematical terminology and notation
(including diagrams and sketches drawn in related exercises), strategies and procedures used to
complete application problems.
Procedures:
Students will:
▪ use the pipe cleaners in demonstrating possible side length combinations for
triangles with a perimeter of 12”.
▪ complete the chart provided, using the Triangle Inequality Theorem and the Pythagorean
Theorem and its converse forms to draw conclusions on triangle type.
▪ complete related exercises (difficult extensions) and illustrate solutions with appropriate
sketches.
▪ show all work/computations that support conclusions drawn throughout the activity.

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
Name: ………………….............
Students’ Directions for Triangle Inequality Rubric Assessment:
♦ Mark your pipe cleaner off in inches by marking the manipulative with a Sharpie
marker at each inch interval (from one to eleven inches).
♦ Bend your pipe cleaner to show the length values indicated in the chart below, then
connect the tips of the pipe cleaner together to form a triangle, if possible (without
disturbing the side lengths chosen). Record whether you could create the triangle in
the chart (yes/no). Perform the indicated computations in the chart.
♦ Continue testing each set of values, fill in the missing combinations to complete the
chart to indicate ALL possible combinations. For those values that DO form a triangle,
apply the Pythagorean Theorem and its converse forms to determine what type of
triangle is formed. Show all computations to support all your conclusions – three
inequalities to support each set that DOES form a triangle, one inequality for each
counterexample, and one inequality or equation that supports your conclusion on type
of triangle formed.
♦ Answer all questions fully and thoroughly; complete related exercises.
♦ EXAMINE THE RUBRIC FIRST as a preview on grading categories and expectations.
☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺
1. Fill in the chart below as each observation is made while attempting to form a triangle
with the specified side lengths:
SIDE
LENGTHS
SUM OF
TWO SIDES
APPLY TRIANGLE
INEQUALITY THEOREM
APPLY PYTHAGOREAN THM
AND ITS CONVERSE FORMS
AB BC AC AB + BC AB + AC BC + AC triangle formed? (Y/N) type of ∆ (or N/A)
1 1 10 2 11 11 NO; 1 + 1 not > 10 N/A (example problem)
1 2 9
1 3 8
1 4 7
1 5 6
2 2 8
2 3 7
2 4 6
2 5 5
3 □ □
3 □ □
4 □ □

2. Computations to support ∆s formed – use of Triangle Inequality Theorem/Pythagorean
Theorem and its converse forms: (clearly label your work)
3. Computations to support values that do NOT determine a ∆ - state theorem applied:
4. Related Exercises (constructed responses) – complete each problem – show all
computations, explain methods, and supply sketches.
:
A. A triangle has a perimeter of
20 cm. List one possible set of side
lengths for this triangle (show why
these lengths “work”).
B. A RIGHT triangle with one side length of
9 cm has a perimeter of 36 cm. List the
other two side lengths (whole numbers)
and show why these measures fit the
description given in the problem.

Triangle Inequality Theorem Assessment Rubric
(ORIGINAL RUBRIC)
Student Name: …………………………………………………………………………………………………
CATEGORY 4 3 2 1
Mathematical Concepts
Explanation shows
complete understanding of
the theorems and converse
forms used to solve the
problem(s).
Explanation shows
substantial understanding
of the theorems and
converse forms used to
solve the problem(s).
Explanation shows some
understanding of the
theorems and converse
forms needed to solve the
problem(s).
Explanation shows very
limited understanding of
the underlying concepts
needed to solve the
problem(s) OR is not
written.
Mathematical Reasoning
Uses complex and refined
mathematical reasoning
through work shown.
Uses effective
mathematical reasoning
through work shown.
Some evidence of
mathematical reasoning.
Little evidence of
mathematical reasoning.
Use of Manipulatives
Student always listens and
follows directions and only
uses manipulatives (pipe
cleaners) as instructed.
Student typically listens
and follows directions and
uses manipulatives as
instructed most of the time.
Student sometimes
listens and follows
directions and uses
manipulatives
appropirately when
reminded.
Student rarely listens
and often "plays" with
the manipulatives
instead of using them as
instructed.
Explanation and checking
Explanation (including
evidence of checking by
use of theorem) is detailed
and clear.
Explanation (including
evidence of checking by
use of theorem) is logical.
Explanation is a little
difficult to understand,
but includes critical
components. Checking is
incomplete.
Explanation is difficult to
understand and is
missing several
components OR was not
included. Checking not
evident.
Completion
All problems are
completed.
All but 1 of the problems
are completed.
All but 2 of the problems
are completed.
Several of the problems
are not completed.
Mathematical Terminology
and Notation, including use
in Diagrams and/or
Sketches
Correct terminology and
notation are always used
(including in sketches),
making it easy to
understand what was done.
notation are usually used
(including in sketches),
making it fairly easy to
notation are used, but it is
sometimes not easy to
understand what was
done (some information
may be missing in
sketches)
There is little use, or a lot
of inappropriate use, of
terminology and
notation.
Strategies and/or
Procedures for Applications
Problems
Typically, uses an efficient
and effective strategy to
solve the problem(s).
Typically, uses an effective
strategy to solve the
problem(s).
Sometimes uses an
effective strategy to solve
problems, but does not do
it consistently.
Rarely uses an effective
strategy to solve
problems.
TOTAL:

REVISED RUBRIC – CCHHEECCKKLLIISSTT AANNDD SSIINNGGLLEE PPOOIINNTT RRUUBBRRIICC CCOOMMBBIINNAATTIIOONN::
I. Yes or No Checklist of Directions – please revise your project until all
responses are YES.
a) ALL problems in the project packet are complete (answered) YES NO
b) ALL problems in the project packet have work shown YES NO
c) Manipulatives (pipe cleaners) are marked off in 1-inch spaces YES NO
d) Manipulatives are glued and displayed clearly, spacing allowed YES NO
e) Sample (five) constructions have side lengths labeled YES NO
f) Sample constructions show use of Δ Inequality Theorem YES NO
g) Sample constructions show use of Converse of Pythagorean
Theorem
YES NO
II. Single-Point Rubric – Use the rubric to revise your project.
NOT YET
(areas that
need work)
PROFICIENT
(performance standards)
EVIDENCE
(how you have met the standard)
ADVANCED
(areas that go beyond the basics)
Mathematical Concepts
Explanation shows complete
understanding of the theorems
and converse forms used to solve
the problem(s).
Mathematical Reasoning
Uses complex and refined
mathematical reasoning through
work shown.
Explanation and Checking
Explanation (including evidence
of checking by use of theorem) is
detailed and clear.
Mathematical Terminology
and Notation, including use
in Diagrams and/or
Sketches
Correct terminology and notation
are always used (including in
sketches), making it easy to
Strategies and/or
Procedures for Applications
Problems
Uses an efficient and effective
strategy to solve the problem(s).

Assessment #2 – selected response assessment
Assessment #2: Triangle Inequality Theorem/Pythagorean Theorem and Converse Forms
Using your results and conclusions from the rubric activity, choose the best response:
1. _____ Which of the following are possible side lengths for a triangle?
A. 5, 9, 15 B. 2, 4, 6 C. 6, 7, 8
2. _____ Two sides of a triangle measure 15 cm and 26 cm. The third side could measure:
A. 17 cm B. 45 cm C. 11 cm
3. _____ The sides of a triangle measure 7 cm, 8 cm, and 9 cm. The triangle is a ? triangle:
A. acute B. obtuse C. right
4. _____ The sides of a triangle measure 5 cm, 5√3 cm, 10 cm. The triangle is a ? triangle:
A. acute B. obtuse C. right
5. _____ In ∆ABC, m∠A = 60, m∠B = k, and m∠C = k + 2. The longest side of ∆ABC is:
A. AB B. BC C. AC
6. _____ In ∆RST, RS = x, ST = x + 1, and RT = x – 1. The smallest angle of ∆RST is:
A. ∠R B. ∠S C. ∠T
7. _____ The base of an isosceles triangle measures 12 cm. The length of the legs could be:
A. 4 cm B. 6 cm C. 8 cm
8. _____ Two sides of a parallelogram measure 10 cm and 12 cm. The diagonals could have lengths of:
A. 6 cm & 10 cm B. 2 cm & 8 cm C. 18 cm & 22 cm
9. _____ In ∆ABC, if AB = BC and AC > BC, then:
A. m∠B < m∠A B. m∠B > m∠C C. m∠B = m∠A
10. ____ In ∆MNP, MN = 8 cm and NP = 10 cm. Which of the following must be true?
A. MP > 2 B. MP > 10 C. MP < 10

(selected response assessment – continued)
TRUE or FALSE. Using your results and conclusions from the rubric activity, answer true or false:
11. _____ A triangle can be formed with sides of lengths 9 cm, 12 cm, and 15 cm.
12. _____ A triangle whose sides measure √3 cm, √4 cm, and √5 cm is an obtuse triangle.
13. _____ A rectangle with sides measuring 7 cm and 24 cm has diagonals that measure 25 cm.
14. _____ In obtuse ∆RST, RT = TS, so it follows that m∠R = m∠S, ∠T is an obtuse angle, and RS > TS.
15. _____ In a triangle in which the lengths of two sides are 5 cm and 9 cm, the length of the third side
is represented by x. It follows that for a triangle to be formed, 5 < x < 9.
16. _____ In ∆ABC, BC > AB and AC < AB. Therefore, m∠B > m∠A > m∠C.
17. _____ In ∆JKM, the side lengths are represented as: JK = n + 2, KM = n, JM = n + 1, where n is a
positive integer. We can conclude that m∠J < 60.
18. _____ In ∆RST, m∠T = 60 and m∠R = 55. It follows that RT > RS.

Assessment #3 – Constructed response assessment
Assessment #3: Triangle Inequality Theorem/Pythagorean Theorem and Converse Forms
1. How many different ∆s are there for which the lengths of the sides are 3, 8, and n, where n
is a whole number and 3 < n < 8? SSHHOOWW WWOORRKK aanndd pprroovviiddee SSKKEETTCCHHEESS::
22.. If the lengths of two sides of an isosceles ∆ are 7 and 15, what is the perimeter of the
triangle? SSKKEETTCCHH ppoossssiibbiilliittiieess aanndd ssuuppppllyy rreeaassoonnss wwhheetthheerr tthheeyy wwoorrkk oorr nnoott::
3. In ∆ABC, BC > AB and AC < AB. SSKKEETTCCHH tthhee ttrriiaannggllee,, iilllluussttrraattee ssiiddee lleennggtthhss,, and arrange the
angles from largest to smallest.

(constructed response assessment – continued)
4. The perimeter of a triangle in which the lengths of all the sides are integers is 21 cm. If the
length of one side of the triangle is 8 cm, what is the shortest possible length of another
side of the triangle? SSHHOOWW WWOORRKK bbyy uussiinngg tthhee ttrriiaannggllee iinneeqquuaalliittyy tthheeoorreemm..
55.. If the integer lengths of the three sides of a triangle are 4, x, and 9, what is the least
possible perimeter of the triangle? SSHHOOWW WWOORRKK bbyy uussiinngg tthhee ttrriiaannggllee iinneeqquuaalliittyy tthheeoorreemm..
66.. If the product of the lengths of the three sides of a triangle is 105, what is a possible
perimeter of the triangle? SSHHOOWW WWOORRKK bbyy uussiinngg tthhee ttrriiaannggllee iinneeqquuaalliittyy tthheeoorreemm..
77.. The sides of a triangle have lengths x, x + 4, and 20. State the values of x for which the
triangle is acute, with the longest side of 20. SSHHOOWW WWOORRKK bbyy uussiinngg tthhee ttrriiaannggllee iinneeqquuaalliittyy
tthheeoorreemm..

(constructed response assessment – continued)
88.. EFGH is a parallelogram with EF = 13, EG = 24, and FH = 10. What kind of parallelogram is
EFGH? UUssee tthhee PPyytthhaaggoorreeaann TThheeoorreemm aanndd iittss ccoonnvveerrssee ffoorrmmss ttoo vveerriiffyy yyoouurr aannsswweerr..
9. Lengths of 7 cm, 8 cm, and 11 cm may represent the sides of a triangle. Use the ttrriiaannggllee
iinneeqquuaalliittyy tthheeoorreemm to determine whether a triangle can be formed, and if so, then use the
PPyytthhaaggoorreeaann TThheeoorreemm aanndd iittss ccoonnvveerrssee ffoorrmmss to determine the type of triangle formed.

Assessment #4 – Journal Response Entry – Constructed response assessment
(Geometry, McDougal-Littell, 2011)

Assessment #5 – Journal Response Entry/Conference – Extended written response
Assessment #5 – Extended written response (as an open-ended verbal response when
conferencing with teacher)
⇛ Each student will have a three-minute conference with the teacher and discuss
how he/she has interpreted both the Triangle Inequality Theorem and the
Pythagorean Theorem and its converses.
1. Look at the snapshot of the Triangle Inequality Theorem below, taken from the
short video viewed in class. Be prepared to explain what ‘is happening’ with the
three sets of colored bars that are below the triangle shown. How are they
related to the property that the theorem states? Write your response (notes for
conference) :

2. Look at the snapshot of the Pythagorean Theorem and its Converse Forms, taken
from the short video viewed in class. Be prepared to explain what ‘is happening’
with the three triangles shown and the highlighted equation and inequalities as
well. What does the property tell us about how to figure out what type of
triangle is formed when we are given three side lengths? Write your response
(notes for conference) :

3. Look at the snapshot of the Pythagorean Theorem and
its Converse Forms,
taken from the short video viewed in class. Be prepared to explain how to
label sides a, b, and c, and how to set up and solve the problems below. Bring
your calculator to the conference. Write your response (notes for conference
– include setting up the problems – you will solve them during conferencing) :
(You can use the space next to each problem below to show work during your conference).
next – survey…

Assessment #6 – Student Survey: Triangle Inequality Theorem & Pythagorean Theorem Converse Forms
Name___________________________________________________________
Please complete survey questions as honestly as you can. These questions are designed to help
teachers better understand students’ comfort level with the concepts learned, as well as their comfort
level with the methods and activities utilized in helping students to learn the material.
1. How comfortable do you feel about your understanding the properties we learned in the Triangle
Inequality Theorem and the Pythagorean Theorem Converses?
(check only one choice)
 very comfortable
(can help others)
 somewhat
comfortable
 somewhat
uncomfortable
 very uncomfortable
(need more help)
2. Which methods listed below did you find most beneficial to you in helping you understand the
concepts and problems that were completed in this unit? (check ALL that apply)
 regular class
lesson (teacher-led)
 Khan Academy
video & quiz
 other video lessons
 team/group work
with discussion
 rubric project
(with pipe cleaners)
 journaling  conferencing  team presentation
3. You may experience various learning and teaching styles in your other core classes. Please check off
any/all that have been utilized in your current classes:
regular
teacher-led
lesson
videolessons
teamor
groupwork
rubric
projects
journalsor
logs
conference
orinterview
withteacher
team
presentations
other
(specify)
English       
Science       
Social Studies       
4. Within the Khan Academy video there is always a quiz/practice. Rate the ‘average’ Khan
presentation – is it normally good enough for you to understand so that you get five correct answers
in a row rather quickly? (check only one choice)
 YES, I normally have
no problems getting
five in a row correct
after viewing the
Khan lesson
 Usually I need to
complete many
problems before
getting five in a
row correct, but I
don’t need to ask
for help
 NO, I usually have to
ask a friend or a
teacher to help me
get the first few
correct before I try
problems on my own
 NO, I normally don’t
pay much attention
to the Khan lesson
because it’s too
confusing – I’d
rather be taught a
in a real-life lesson
O V E R ⇛

5. When it comes time for the final exam, which skills listed do you think you will need to review?
(check all that apply)
 squaring radicals
 showing (proving)
that a Δ exists
 finding a missing
length
(in forming a Δ)
 using a converse
form of the
Pythagorean Thm to
determine type of Δ
 stating Pythagorean
triples
 analyzing
angle/side
relationships in Δs
 perimeter of Δs
problems with side
lengths missing
 using Algebra to find
missing angles or
missing sides in a Δ
6. Please state any other concerns or suggestions that you may have at this time:
THANK YOU for your honest responses. Your input will help make the class more beneficial to you!

Triangle Inequality Theorem: Activities and Assessment Methods

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