EDG544 Assessment Project (Template)

Name: Marianne McFadden Subject Area: Mathematics – Geometry (non-honors) Grade Level: Grades 8 – 11

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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 ◄◄

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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.

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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

USE OF DATA/RESULTS:

- 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)

Performance Skills

selected response

(multiple choice; true

and false)

Questions relating to using

theorems/converse forms without

manipulatives

(#1, 11, 12)

Product Learning

(#2, 3, 4, 7, 8, 13, 15)

USE OF DATA/RESULTS:

- 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)

Product Learning

(#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)

USE OF DATA/RESULTS:

- 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

USE OF DATA/RESULTS:

- 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)

Performance Skills

(#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

USE OF DATA/RESULTS:

- 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 A B .

Using the figure to the left, state the angle that is opposite the

indicated sides:

5. ________ C A 6. ________ B C

Using the figure to the left, state the following:

In ∆CDB, find the angle opposite of:

7. ________ D B 8. ________ C B

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, A C 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 D E and E F because these sides form ∠E.

Using ∆ABC above, find the angle include between:

14. ________ A C and A B 15. ________ B C and A C

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. ________ D C and C B 19. ________ D B and C D

▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪

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.

Correct terminology and notation are usually used (including in sketches), making it fairly easy to

understand what was done.

Correct terminology and 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 understand what was done.

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. A B B. B C C. A C

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:

regu

lar

teac

he

r-le

d

less

on

vid

eo

less

on

s

team

or

gro

up

wo

rk

rub

ric

pro

ject

s

jou

rnal

s o

r

logs

con

fere

nce

or

inte

rvie

w

wit

h t

eac

he

r

team

p

rese

nta

tio

ns

oth

er

(sp

eci

fy)

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!