Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Overview of Instructional Task:
Applying Angle Theorems—Four Pentagons
This task helps students develop strategies for solving problems by applying polygon angle theorems. In this
lesson, students work individually on the instructional task, Four Pentagons. In the subsequent lesson, students
work on the lesson task, The Pentagon Problem. The lesson task is similar to the instructional task and provides
students with an opportunity to explore important ideas that will inform their revisions of the Four Pentagons
task.
Task should be implemented after students have begun work on Chapter 5, but before complete mastery (after
Lesson 5.3, November 29–December 2, 2011).
One or Two Days Before Lesson
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Students work independently on instructional task, Four Pentagons task (page 2) (15 minutes).
Provide calculators if requested.
Collect student work.
Analyze responses and write questions on each student paper. Do not score at this time.
Lesson
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Students work in small groups to complete the problem using four student methods (pages 3–5) (20 minutes).
Students work with same small groups to analyze sample solutions produced by other students (pages 16–19)
(15 minutes).
Conduct whole class discussion of sample solutions (15 minutes).
Students individually revisit and revise their original work on instructional task, Four Pentagons (10–15
minutes at end of lesson or during next class).
After Lesson
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Analyze student responses to identify next instructional steps.
Task may be graded if desired (see DPS Rubric, page 21).
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
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Instructional Task: Applying Angle Theorems—Four Pentagons
Name:
Geometry, Semester 1
Date:
Four Pentagons
Show all work and explain your thinking. Use the back of this paper
as necessary.
This diagram is made up of four regular pentagons that are all the
same size.
1. Find the measure of angle AEJ. Show your calculations and explain your reasons.
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2. Find the measure of angle EJF. Explain your reasons and show how you figured it out.
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3. Find the measure of angle KJM. Explain how you figured it out.
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Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
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Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
The Pentagon Problem
Mrs. Morgan wrote this problem on the board.
This pentagon has three equal sides at the top
and two equal sides at the bottom.
Three of the angles have a measure of 130°.
Figure out the measure of the angles marked x
and explain your reasoning.
Diagram is not accurately drawn.
Four students in Mrs. Morgan’s class came up with different methods to answer this problem. Their methods are shown below.
Use each student’s method to calculate the measure of angle x. Write all your reasoning in detail.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
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Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
1. Annabel drew a line down the middle of the pentagon.
She calculated the measure of x in one of the
quadrilaterals she made.
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2. Brian used the exterior angles of the pentagon to figure
out the measure of x.
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Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
4
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
3. Carlos drew a line that divided the pentagon into
a trapezoid and a triangle.
Angle x has also been cut into two parts, so he labeled
the parts a and b.
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4. Diana divided the pentagon into three triangles to
calculate the measure of x.
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Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
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Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Instructional Task:
Applying Angle Theorems—Four Pentagons
Mathematical Goals
This instructional task helps assess how well students:
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Solve problems related to measures of interior angles of polygons.
Solve problems related to measures of exterior angles of polygons.
Common Core State Standards
This instructional task emphasizes the following Standards for Mathematical Practice:
3. Construct viable arguments and critique others’ reasoning.
7. Look for and make use of structure.
This instructional task also asks students to select and apply mathematical content from the Common Core State
Standards.
G-SRT.9: Prove theorems about lines and angles.
G-SRT.10: Prove theorems about triangles.
G-SRT.11: Prove theorems about parallelograms.
Required Materials
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Copies of instructional task, Four Pentagons, for each student (page 2)
Copy of lesson task, The Pentagon Problem, for students (pages 3–5)
Copies of student work samples for each small group (pages 16–19)
Slides of Instructions for The Pentagon Problem (pages 13–14)
Slide of Instructions for Evaluating Student Work Samples (page 15)
Time
Times given are only approximate. Exact timings depend on class needs.
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One to two days before lesson: 15 minutes
Lesson: 60 minutes
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
6
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Before Lesson
Instructional Task: Four Pentagons (15 minutes)
Have students do this task in class a day or more before the lesson task, so you can assess the work, find out the
kinds of difficulties students are having, and more effectively focus the follow-up lesson.
Distribute copies of the instructional task, Four Pentagons (page 2).
Introduce the task briefly and help the class understand the task
and its context.
Read the questions and try to answer them as carefully
as you can. Show all your work so I can understand your
reasoning. Use another paper if you need more room.
In addition to trying to solve the task, I want to see if you
can present your work in an organized and clear manner.
It is important that students answer the questions without
assistance, as far as possible.
Provide calculators, if needed or requested.
Ask students to attempt the task on their own, without discussion.
Don't worry if you cannot understand everything, because
there will be a lesson on this material [tomorrow] that will
help. By the end of the next lesson, you should expect to
be more confident when answering questions like these.
Provide Student Feedback
Collect students’ responses to the task. Make some notes on what their work reveals about their current levels
of understanding and their different problem-solving approaches.
Do not score students’ work. Research shows that it is counterproductive, as it encourages students to compare
their scores and distracts their attention from what they can do to improve their mathematics.
Instead, help students to make further progress by summarizing their difficulties as a series of questions, such as
the suggestions that follow. Write questions on each student’s work that will advance his or her thinking. You
may also note students with particular issues, so you can monitor their work during the lesson.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
7
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Common Issues
Suggested Questions and Prompts
Student has difficulty getting started.
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Student writes little in response to any questions.
Student works unsystematically.
Student presents work poorly.
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Student makes arithmetic errors.
For example: Student writes, “Angle
EJF=180°−144=46°.”
Student uses incorrect formula.
For example: Student does not identify correct
formula to use to find interior angle of pentagons in
question 1.
Student produces partially correct solution.
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What can you tell me about the task?
Write what you know about this diagram.
How might that information be useful?
What else can you calculate?
Can you organize your work in a systematic way?
Would labeling your work help make it clearer?
Would someone unfamiliar with your type of
solution easily understand your work?
Have you explained how you arrived at your
answer?
Could you write an explanation to next year’s
students of how to do this task?
How can you be sure your answer is correct?
Does your answer make sense?
Where can you find the correct formula for the
interior angle of a regular pentagon?
What does n stand for in this formula?
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You have given an answer of [216°]. Which angle is
this one on the diagram? What do you need to do
to complete your solution?
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Angles in parallelograms are supplementary, but how
do you know that this figure is a parallelogram?
Student provides poor reasoning.
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For example: Student calculates using a conjecture but
does not state what the conjecture is.
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Student produces full solution.
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How do you know that this calculation is the
correct one to perform?
Would someone reading your solution understand
why your answer is correct?
Find another way to solve each part of the Four
Pentagons problem.
For example: Student does not follow through on the
method she or he has written.
Or: Student calculates 540° but does not find interior
angle.
Or: Student calculates 108° or 216° but does not find
angle AEJ.
Student uses unjustified assumptions.
For example: Student argues that supplementary
angles sum to 180° without first establishing that the
figure is a rhombus.
Student provides full and well-reasoned solution and
has justified all assumptions.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
8
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Suggested Lesson Outline
If you have a short class period or you find the lesson is progressing more slowly than anticipated, we suggest
ending the lesson after the first collaborative activity and continuing in a second lesson.
Collaborative Problem Solving: The Pentagon Problem (20 minutes)
Organize students into small groups of two or three. Give each
group a copy of The Pentagon Problem (pages 3–5). Display the
slides, Instructions for The Pentagon Problem (pages 13–14).
Introduce the task and explain what you are asking students to do.
Mrs. Morgan is a teacher in another school. She wrote this
problem on the board for her students. I’m giving you some
work written by four of her students.The students all used
different methods to solve the problem. First, take a few
minutes to individually complete The Pentagon Problem.
Then, I want you to use each student’s method in turn to
solve The Pentagon Problem.To get started, choose one
method and work together to produce a solution. Make
sure everyone in your group understands how that method works. Then move to the next method.
Write all your reasoning in detail and make sure you justify every step.
As students work, note student difficulties and support student problem solving.
Note Different Student Approaches to Task
Look for students’ difficulties with particular solution methods. Which solution method(s) do they find most
difficult to interpret and use? What is it that they find difficult? Notice also ways they justify and explain to each
other. Do they justify assumptions? Do they explain all their calculations in reference to conjectures and
definitions? Use this information to focus a whole class discussion towards the end of the lesson.
Support Student Problem Solving
Try not to focus on numerical procedures for deriving answers. Instead, ask students to explain their
interpretations and uses of different methods. Raise questions about their assumptions and prompt for
explanations based on angle theorems to encourage precision in students’ reasoning. Refer them to geometric
definitions and conjectures as needed.
Collaborative Analysis of Student Work Samples (15 minutes)
As students complete their solutions, give each small group of students copies of the student work samples
(pages 16–19) and ask for written comments. Display the slide, Instructions for Evaluating Student Work
Samples (page 15). This step gives students the opportunity to evaluate a variety of possible approaches to the
task, without providing a complete solution strategy.
Four students in another class used Annabel, Brian, Carlos, and Diane’s methods to solve the problem like
you just did. Here are copies of the other students’ work.None of this work is perfect! For each student’s
solution, explain if the reasoning is correct and complete. Correct the method when necessary. Use the
method to calculate the measure of the missing angle, giving detailed reasons for all your answers.
Each student work sample poses specific questions for students to answer. In addition to these questions, you
could ask students to evaluate and compare responses. To help them do more than check if the answer is
correct, you may want to ask the following questions (page 20).
• How did this student organize his or her work?
• What mistakes have been made?
• What misconceptions do you think this student has?
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
9
Instructional Task: Applying Angle Theorems—Four Pentagons
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Geometry, Semester 1
What isn’t clear?
What questions would you like to ask this student?
In what ways might the work be improved?
Every group may not have enough time to work through all student work sample questions. If so, be selective
about what you hand out.
During small group work, support students as before. Note similarities and differences between students’
approaches during small group work and student work sample approaches. Also check which methods students
have difficulties understanding to focus the next activity, a whole class discussion.
Whole Class Discussion: Compare Different Approaches (15 minutes)
Organize a whole class discussion to consider different approaches used in the student work samples (pages
16–19). Focus the discussion on those parts of the small group tasks that students found difficult. Ask students
to compare different solution methods.
Using your understanding of your students’ difficulties from the instructional task and their work during the
lesson, choose one student work sample to discuss. Ask one group to present its analysis of that response.
Ask other students for comments and reactions.
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[Celia] What went wrong in Megan’s solution?
Why did Brian draw that line?
Can you explain what assumption Katerina made? Was it a correct assumption?
[Trevor] Can you explain it another way?
Then look at another solution method.
Finally, compare methods.
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Which student’s work provided the most complete reasoning?
Which student’s work was most difficult to understand?
Which approach did you like best? Why?
Which approach did you find most difficult to understand? Why?
The intention is that students begin to realize the power of using different methods to solve the same problem
and appreciate the need for, and nature of, adequate reasons for each assertion.
Individually Review Original Solutions to Task (10–15 minutes)
Return students’ work on the instructional task, Four Pentagons. Distribute blank copies of the Four Pentagons
instructional task.
Ask students to read their responses, remembering what they learned during this lesson. Ask students to revise
their original responses to the task.
Look at your original responses and think about what you learned in this lesson.Using what you learned,
try to improve your work.
If time runs out during the lesson, you could do a follow-up lesson.
Solutions
Instructional Task: Four Pentagons
1. The measure of angle AEJ is 144°.
2. The measure of angle EJF is 36°.
3. The measure of angle KJM is 108°.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
10
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Lesson Task: The Pentagon Problem
Each method gives a way to calculate the measure of angle x, 75°. Each method uses different definitions and
angle properties in the explanation.
1. Annabel’s method
Some students may not understand the need to justify the assumption
that the line “down the middle of the pentagon” bisects the130° angle
at the base of the pentagon.
The construction line divides AC into segments of equal length.
So AB = BC.
AF = CD is given.
Angle BAF is congruent to angle BCD.
So by SAS, triangles ABF and BCD are congruent.
Triangle BFE is congruent to triangle BDE by SSS.
So angle FEB = angle BED =
130
= 65°.
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To show that the two quadrilaterals ABEF and BCDE are congruent:
The sides are all congruent as BA= BC, AF = CD, FE = DE, and BE is common to both quadrilaterals.
The angle between sides AB and AF is congruent to the angle between sides BC and CD.
The angle between sides AF and FE is congruent to the angle between sides and DE.
So the quadrilaterals are congruent.
The figure is therefore symmetrical. So angle ABE = angle CBE = 90°.
Since the sum of the angles in a quadrilateral is 360°, x = 360 – (90 + 130 + 65) = 75°
2. Carlos’s method
Some students may make the false assumption that all the exterior
angles are congruent.
The sum of an interior and an exterior angle is 180°.
Three of the angles of the pentagon are known; all three are 130°.
The exterior angle for each of these interior angles is 180 – 130 = 50°.
The sum of the exterior angles of a polygon is 360°.
360 – 3 × 50 = 360 – 150 = 210°.
This is the sum of the two missing exterior angles.
The two missing interior angles are congruent.
x = 180 –
1
× 210 = 180 – 105 = 75°.
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3. Brian’s method
The pentagon is divided into a quadrilateral and a triangle.
In trials, some students did not understand the need to justify the claim
that the quadrilateral is a trapezoid, and others did not understand the
need to show that both triangle and trapezoid are isosceles.
The triangle is isosceles because it has two congruent sides. So the angles
marked a are congruent. So the angles marked b = x − a are also
congruent to each other.
The quadrilateral is an isosceles trapezoid because the two slant sides are
congruent and meet the horizontal side at congruent angles.
It follows that the base is parallel to the top, and angles marked b are also congruent.
The angles in a triangle sum to 180°.
2a= 180 – 130 = 50
a = 25°
The angles in a quadrilateral sum to 360°.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
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Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
2b= 360 – 2 × 130 = 100
b = 50°
Alternatively, since the top and base of the trapezoid are parallel, the angles b and 130° are supplementary,
and b= 180 – 130 = 50°
4. Diane’s method
Some students may rely on the appearance of the diagram, assuming the
three triangles are all isosceles.
Diane shows the pentagon divided into three triangles. The sum of the
angles in any triangle is 180°. The sum of the angles in the pentagon is
thus 180 × 3 = 540°.
The three known angles are all 130°.
The two unknown angles are congruent.
2x = 540 – 3 × 130 = 150°
x = 75°
The outer triangles are not isosceles.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
12
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Instructions for the Pentagon Problem
Mrs. Morgan wrote a problem on the board.
“This pentagon has three equal sides at the top
and two equal sides at the bottom.
Three of the angles have a measure of 130°.
Figure out the measure of the angles marked x
and explain your reasoning.”
Diagram is not drawn accurately.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
13
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Instructions for The Pentagon Problem (cont.)
• Four students in Mrs. Morgan’s class came up with different
methods to answer this problem.
• Their methods are shown on the worksheets.
• Use each student’s method to calculate the measure
of angle x.
• Write all your reasoning in detail.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
14
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Evaluating Student Work Samples
Four students answered The Pentagon Problem using
Annabel, Carlos, Brian, and Diana’s methods.
For each piece of work,
• Explain whether the student’s reasoning is correct
and complete.
• Correct the solution if necessary.
• Use the method to calculate the measure of angle x.
• Write all your reasoning in detail.
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
15
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Sample 1: Erasmus
Erasmus used Annabel’s method.
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Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
16
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Sample 2: Tomas
Tomas used Brian’s method.
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Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
17
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Sample 3: Katerina
Katerina used Carlos’s method.
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Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
18
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Sample 4: Megan
Megan used Diane’s method.
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Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
19
Instructional Task: Applying Angle Theorems—Four Pentagons
Geometry, Semester 1
Evaluating Student Work Samples
• How did this student organize his or her work?
• What mistakes have been made?
• What misconceptions do you think this student has?
• What isn’t clear?
• What questions would you like to ask this student?
• In what ways might the work be improved?
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
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Instructional Task: Applying Angle Theorems—Four Pentagons
Task adapted from MARS Shell Center, University of Nottingham and UC Berkeley, © 2011 MARS University of Nottingham
Geometry, Semester 1
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