CFA 2.0 DESIGN TEMPLATEimages.pcmac.org/.../Documents/5.1_Improving_Instruction_CFA_FLA.pdfCFA 2.0 DESIGN TEMPLATE. GRADE AND CONTENT AREA Eleventh Grade: United States History II:

CFA 2.0 DESIGN TEMPLATE

GRADE AND CONTENT AREA Eleventh Grade: United States History II: The Industrial Revolution to the Present

NAME OF UNIT Civil Rights 4th Quarter

TIME FRAME (UNIT + BRIDGE) 2 to 3 weeks

AUTHORS

PRIORITY STANDARDS

14: Trace events of the modern Civil Rights Movement from post-World War II to 1970 that resulted in social and economic changes, including Montgomery Bus Boycott, the desegregation of Little Rock Central High School, the March on Washington, Freedom Rides, the Sixteenth Baptist Church bombing. 15: Describe changing social and cultural conditions in the United States during the 1950s, 1960s, and 1970s.

SUPPORTING STANDARDS

14.1: Tracing the federal government’s involvement in the modern Civil Rights movement, including the abolition of the poll tax, the nationalization of state militias, Brown versus Board of Education in 1954, the Civil Rights Acts of 1957 and 1964, and the Voting Rights Act of 1965. 14.2: Explaining contribution of individuals and groups to the modern Civil Rights Movement, including Martin Luther King, Jr.; James Meredith; Medgar Evans; Thurgood Marshall; the Southern Christian Leadership Conference (SCLC); the Student Nonviolent Coordinating Committee (SNCC); The Congress of Racial Equality; the National Association for the Advancement of Colored People (NAACP); and the civil rights foot soldiers.

“UNWRAPPED” PRIORITY

STANDARDS

14: Trace events of the modern Civil Rights Movement from post-World War II to 1970 that resulted in social and economic changes, including Montgomery Bus Boycott, the desegregation of Little Rock Central High School, the March on Washington, Freedom Rides, the Sixteenth Baptist Church bombing. 15: Describe changing social and cultural conditions in the United States during the 1950s, 1960s, and 1970s.

“UNWRAPPED” PRIORITY STANDARDS GRAPHIC ORGANIZER

“Unwrapped” Priority Standards CONCEPTS

“Unwrapped” Priority Standards SKILLS

Levels of Cognitive Rigor

BLOOM’S DOK

Trace events of the modern Civil Rights Movement from post-World War II to 1970.

Abolition of the poll tax

Nationalization of state militias

Impact of court cases and legislation (Brown versus Board of Education, Civil Rights Acts of 1957 and 1964 & Voting Rights Act of 1965).

Trace (events of the modern Civil Rights Movement from post-World War II to 1970).

Understanding

3

Era of social change

Women fight for equality

Latinos, Asian Americans, and Native Americans seek equality

Culture and counter-culture

Describe (changing social and cultural conditions in the United States from 1950s-1970s).

Understanding

3

BIG IDEAS

1. After World War II, African Americans began winning important victories in the battle for civil rights. An awareness of our history helps us to develop tolerance today.

2. During post-World War II Latinos, Asian Americans, Native Americans and women began effective equal rights movement of their own. Understanding the struggles of select groups allows the awareness of cultural differences and promotes an appreciation of those differences. An awareness of bias and propaganda effect on our daily lives.

ESSENTIAL QUESTIONS

1. In what ways did African Americans fight discrimination during the civil rights era? How does developing an awareness of these events impact my life today?

2. Why did social protests and calls for change sweep across the United States in the 1960s? What changes might protestors be seeking today in the United States? How do these protests from across the country affect every day citizens?

UNIT LEARNING INTENTIONS WRITTEN AS STUDENT SUCCESS CRITERIA

STUDENT SUCCESS CRITERIA

Essential Questions “Unwrapped” Priority Standards Unit Vocabulary

Respond correctly with your Big Ideas to the unit’s

Essential Questions:

Demonstrate your ability to do each of the following:

Define and use the unit vocabulary terms

appropriately:

In what ways did African Americans fight discrimination during the civil rights era? Why did social protests and calls for change sweep across the United States in the 1960s?

14: Trace events of the modern Civil Rights Movement from post-World War II to 1970 that resulted in social and economic changes, including: Montgomery Bus Boycott, the desegregation of Little Rock Central High School, the March on Washington, Freedom Rides, the Sixteenth Baptist Church bombing.

15: Describe changing social and cultural conditions in the United States during the 1950s, 1960s, and 1970s.

Thurgood Marshall

Brown v Board of Education

Sit-in

Civil Rights Act of 1964

Voting Rights Act of 1965

American Indian Movement (AIM)

United Farm Worker Organizing Committee

National Organization for Women (NOW)

counterculture

Optional Student Friendly Version of Priority Standards: Explain how civil rights protest succeeded in battling segregation. Identify social and cultural changes that took place in the 1950s, 1960s, and 1970s.

LEARNING PROGRESSIONS SEQUENCED TO UNIT LEARNING INTENTIONS AND STUDENT SUCCESS CRITERIA and QUICK PROGRESS CHECKS MATCHED TO LEARNING PROGRESSIONS

Subskill Enabling Knowledge Quick Progress Check

1. identify early efforts to end segregation Students complete and exit slip

2. identify events leading to the Birmingham protest, 1963 Student complete graphic organizer for independent practice.

3. compare and contrast ideals and lifestyle of the counterculture

Students complete a Venn diagram

(add a mini-debate)

SELECTED-RESPONSE QUESTIONS (MULTIPLE CHOICE ONLY) WITH ANSWER KEY AND DISTRACTOR ANALYSIS

1. 14.0 Trace events of the modern Civil Rights Movement from post-World War II to 1970 that resulted in social and economic changes, including Montgomery Bus Boycott, the desegregation of Little Rock Central High School, the March on Washington, Freedom Rides, the Sixteenth Baptist Church bombing. 1. The initial fate of the Freedom Riders demonstrated that A. Southerners had largely accepted Northern orders to integrate bus stations and other public facilities. B. state governments were at the forefront in the enforcement of civil right laws. C. television news broadcasts had a powerful hold on the American public. D. by 1961 the federal government was committed to vigorously protecting the civil rights of all citizens.

Distractor Analysis: A. Southerners did not accept federal government orders to end segregation in public facilities. B. Southern state governments did not support integration of the races. C. Correct: The images of burned buses and beaten Freedom Riders horrified many Americans. D. At this point, neither the federal nor the state governments protected the rights of Freedom Riders.

2. Question and Answer Choices (with correct answer underlined): 2. Some Northern blacks were attracted to the call for “black power” for all the following reasons except A. Martin Luther King and others in the civil rights movement seemed more interested in improving the position of Southern blacks B. Ghetto sections of Northern cities remained poor, and many residents there felt little hope. C. Malcolm X and Stokely Carmichael evoked powerful images of black pride. D. Vast numbers of Northern blacks had joined the Nation of Islam.

Distractor Analysis: A. The Civil Rights Movement wanted to bring an end to the Jim Crow and segregation in the deep south. B. The plight of Northern blacks often went overlooked by the participants of the Civil Rights Movement. C. Malcolm X and Stokely Carmichael rhetoric focused on a renewal of pride in black people and their communities. D. Correct: All of the other reasons caused some Northern blacks to abandon the Civil Rights Movement call for integration. Only a small proportion of blacks ever joined the Nation of Islam.

3. Question and Answer Choices (with correct answer underlined): 15: Describe changing social and cultural conditions in the United States during the 1950s, 1960s, and 1970s. 1. Highlights for feminist leaders of this era (1960s)included all of the following except A. the founding of Ms. B. the formation NOW C. the drive for passage of the Equal Rights Amendment D. the increased awareness of “women issues” in society

Distractor Analysis: A. Gloria Steinem founded the feminist magazine Ms. B. In 1966, NOW (National Organization for Women) which was dedicated to getting equal pay and protesting stereotypical images of women in the media. C. Correct: After a long struggle, the drive to get the ERA into the Constitution was finally abandoned when it became obvious that not enough state legislatures would ever pass it. D. Women support groups became common on both college campuses and in suburban communities.

SHORT CONSTRUCTED-RESPONSE QUESTIONS WITH SCORING GUIDES

1. How were the Supreme Court cases Plessy v. Ferguson and Brown v. Board of Education of Topeka related?

Scoring Guide: Scoring Guide: Achieves Success Criteria

Demonstrates all

Include two or three sentence explanation.

Organize ideas logically.

Use content vocabulary.

Answers will vary Developing Success Criteria

Demonstrates two of the Achieves Success Criteria. Beginning Success Criteria Demonstrates fewer than two of the Achieved Success Criteria.

2. After reading an excerpt from “I Have a Dream” by Dr. Martin Luther King, Jr. What criticism does Dr. King level at American Society?


Demonstrates all




Answers will vary.

Developing Success Criteria


3.

In what ways did aspects of the counterculture impact mainstream society?


Demonstrates all




Answers will vary; at a minimum cite three sources.



ACCOMPANYING COMMENTARY OR SOLUTION STATEMENT WITH KEY ELEMENTS OF CORRECT RESPONSE

1. The Legislative Branch of the federal government has been the cornerstone of desegregating American society.

2. “I Have a Dream” by Dr. Martin Luther King, Jr. criticizes America’s for its appeasement of “vicious

racist” and of a governor who ignores federal law in order to support racism and violence targeting

African Americans.

3. Pop art became popular among many Americans, including rock n’ roll which became of cultural phenomenon affecting mainstream America.

EXTENDED-RESPONSE QUESTION WITH SCORING GUIDE

1. Do you think the nonviolence used by civil rights activist was a good tactic? If more violence had been incorporated what do you think the outcome would have been for the United States? Discuss the opposing viewpoints of Malcolm X and Dr. Martin Luther King, Jr., and elaborate on their strategies for gaining rights for black citizens.


Demonstrates all

Include two or three sentence explanation; include strategies from both sides.



Answers will vary.



2. What assumptions and beliefs do you think guided the fierce opposition to the civil rights movement in the south? Support your claim (thesis) with evidence by incorporating specific examples. (people and events)


Demonstrates all




Answers will vary.



3. What role did the counterculture and antiwar movement play in helping Richard Nixon win the presidency? What were the shared values of mainstream America during the decades following WWII that led to national unrest in the 1950s, 1960s, and 1970s?


Demonstrates all




Answers will vary. Developing Success Criteria


ACCOMPANYING COMMENTARY OR SOLUTION STATEMENT WITH KEY ELEMENTS OF CORRECT RESPONSE

1. Those who opposed segregation became powerless through the non-violence philosophy of the Civil Rights Movement.

2. Jim Crow was a powerful weapon used by racist in the South that perpetuated the use of violence to control African Americans. It promoted segregation of the races reinforced by bigotry and violence and feeling of superiority towards African Americans.

3. The formation of different organizations was one of the primary tactics used by members of the counterculture movement to promote social and economic changes in America.

4. Shared values often included specific roles or models for work and home that often impacted on the daily lives of both men and women regardless of race and income. * religion, role of women, technology, social , media, politics, music and books, birth control, abortion, education, home, business, etc.

ESSENTIAL QUESTIONS WITH SCORING GUIDE

1. In what ways did African Americans fight discrimination during the civil rights era?


Demonstrates all



Use content vocabulary. * give a range

Answers will vary Developing Success Criteria


2. Why did social protests and calls for change sweep across the United States in the 1960s?


Demonstrates all

Include two or three sentence explanation that supports your response.





3.


Demonstrates all






Pre-Test: Civil Rights Movement:

1. The first of the confrontations on the Edmund Pettus Bridge during the march on Selma became known as a. “Black Friday b. “Selma Saturday c. “ Freedom Summer” d. “Bloody Sunday” 2. What is an argument that people used to prevent affirmative action? a. It would eventually end segregation. b. It would remedy the legacy of racial discrimination. c. It would provide equal opportunities to all. d. It would violate the goal of creating a colorblind society 3. Which civil rights organization won a number of important court cases against segregation in the 1950s? a. SCLC b. CORE c. MIA d. NAACP 4. Why were many Americans surprised when President Johnson supported civil rights? a. As a congressman, he had authored “The Southern Manifesto.” b. He had never been a strong supporter of Southern traditions. c. He was a Southerner with an undistinguished record on racial matters. d. He had openly supported segregation 5.The Civil Rights Act of 1964 forbids discrimination in employment on the basis of a. education b. race c. age d. gender 6. The counterculture of the 1960s valued and promoted a. corporate and business influence b. personal freedom and individuality c. beautifying America d. classical music 7. The Latino population in the United States is made up of people whose family origins are in a. Africa b. Spain c. Latin America d. Asia 8. Which of these were the results of the women’s movement of the 1960s?

a. The Equal Rights Amendment became law. b. Women began earning more than men in most jobs. c. Women won the right to end an unwanted pregnancy. d. Women were no longer allowed to join the military. 9. The American Indian Movement worked to: a. secure citizenship rights for Native Americans b. gain Native American self-government and to regain traditional Native American lands c. convince Native American children to accept white culture d. sell Native American lands to large industries 10. The word feminism describes the theory that a. women are superior to men b. women should have equal rights with men c. women should focus on traditional roles d. women should not work outside the home

CONCEPT DEVELOPMENT

Mathematics Assessment Project

CLASSROOM CHALLENGES A Formative Assessment Lesson

Applying Angle

Theorems

Mathematics Assessment Resource Service

University of Nottingham & UC Berkeley

For more details, visit: http://map.mathshell.org © 2015 MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at http://creativecommons.org/licenses/by-nc-nd/3.0/ - all other rights reserved

Teacher guide Applying Angle Theorems T-1

Applying Angle Theorems

MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, it will support you in identifying and helping students who have the following difficulties:

• Solving problems relating to using the measures of the interior angles of polygons. • Solving problems relating to using the measures of the exterior angles of polygons.

COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics:

7.G: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics, with a particular emphasis on Practices 3, 6, and 7:

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure.

INTRODUCTION The lesson unit is structured in the following way:

• Before the lesson, students work individually to complete an assessment task designed to reveal their current understanding and difficulties.

• During the lesson, students work in pairs or threes on a collaborative discussion task. They are shown four methods for solving an angle problem and work together to complete the problem using each of the methods in turn. As they do this, they justify their work to each other.

• Working in the same small groups, students analyze sample solutions to the same angle problem produced by students from another class. They identify errors and follow reasoning in the sample solutions.

• There is then a whole-class discussion, in which students explain the reasoning in the sample solutions and compare the methods.

• Finally, students return to their original task and try to improve their own responses.

MATERIALS REQUIRED • Each student will need two copies of the assessment task Four Pentagons and a copy of the

lesson task The Pentagon Problem. • Each small group of students will need a copy of each of the Sample Responses to Discuss and a

copy of the Geometrical Definitions and Properties sheet.

TIME NEEDED 15 minutes before the lesson, a 60-minute lesson, and 10 minutes in a follow-up lesson. All timings are approximate. Exact timings will depend on the needs of the class.


BEFORE THE LESSON

Assessment task: Four Pentagons (15 minutes) Have the students do this task in class or for homework a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. You should then be able to target your help more effectively in the subsequent lesson.

Give out the assessment task Four Pentagons. Introduce the task briefly and help students to understand the problem and its context.

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.

It is important that, as far as possible, students are allowed to answer the questions without assistance.

Students who sit together often produce similar answers and then when they come to compare their work, they have little to discuss. For this reason, we suggest that when students do the task individually, you ask them to move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their usual seats. Experience has shown that this produces more profitable discussions.

Assessing students’ responses Collect students’ responses to the task and read through their papers. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches. The purpose of this is to forewarn you of issues that will arise during the lesson itself, so that you may prepare carefully.

We suggest that you do not score students’ work. The research shows that this will be counterproductive, as it will encourage students to compare scores and distract 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. Some suggestions for these are given in the Common issues table on the next page. These have been drawn from common difficulties observed in trials of this lesson unit.

We suggest you make a list of your own questions, based on your students’ work. We recommend you either: • write one or two questions on each student’s work, or • give each student a printed version of your list of questions and highlight the appropriate

questions for individual students.

Applying Angle Theorems Student Materials Beta Version

© 2012 MARS University of Nottingham S-1

Four Pentagons 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.

2. Find the measure of angle EJF.

Explain your reasons and show how you figured it out.

3. Find the measure of angle KJM.

Explain how you figured it out.


If you do not have time to do this, you could select a few questions that will be of help to the majority of students and write these on the board when you return the work to the students in the follow-up lesson.

Common issues: Suggested questions and prompts:

Has difficulty getting started The student writes little in response to any of the questions.

• Write what you know about this diagram. • How might that information be useful? • What else can you calculate?

Makes arithmetic errors For example: The student writes, “Angle EJF = 180° − 144° = 46°.”

• How can you be sure your solution is correct?

Uses an incorrect formula For example: The student does not identify the correct formula to use to find the interior angle of a pentagon (Q1).

• Find the correct formula for the interior angle of a regular pentagon.

• What does n stand for in this formula?

Produces a partially correct solution For example: The student does not follow through the method s/he has written down.

Or: The student calculates 540° but does not find interior angle.

Or: The student calculates 108° or 216° but does not find angle AEJ.

• You have given an answer of [216°]. Which angle is this on the diagram?

• What do you need to do to complete your solution?

Uses unjustified assumptions

For example: The student argues that supplementary angles sum to 180°, without first establishing that the figure is a rhombus.

• The angles in a parallelogram are supplementary, but how do you know that this is a parallelogram?

Provides poor reasoning For example: The student calculates using a theorem but does not state what the theorem is.

• How do you know that this is the correct calculation to perform?

• Would someone reading your solution understand why your solution is correct?

Produces a full solution

The student provides a full and well-reasoned solution and has justified all assumptions.

• Can you find another way of solving each part of the Four Pentagons problem?


SUGGESTED LESSON OUTLINE

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 and a copy of the Geometrical Definitions and Properties sheet.

Display Slide P-1 Instructions for The Pentagon Problem (1). 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.

I want you to use each student’s method in turn to solve The Pentagon Problem.

Display Slide P-2 Instructions for The Pentagon Problem (2).

To get started, choose one of the methods and work together to produce a solution. Make sure everyone in your group understands how that method works. Then move on to the next method.

Write all your reasoning in detail and make sure you justify every step.

As students work you have two tasks: to note student difficulties and to support student problem solving.

Note student difficulties Look for difficulties students have 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 the ways they justify and explain to each other. Do they justify assumptions? Do they explain all their calculations with reference to theorems and definitions?

You can use this information to focus the whole-class discussion at the end of the lesson.

Student materials Applying Angle Theorems S-2 © 2014 MARS, Shell Center, University of Nottingham

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 for answering this problem.

Use each student’s method to calculate the measure of angle x.

Write all your reasoning in detail.

Use the Geometrical Definitions and Properties sheet to help.


1. Annabel drew a line down the middle of the pentagon.

She calculated the measure of x in one of the quadrilaterals she had made.

2. Carlos used the exterior angles of the pentagon to figure out the measure of x.






3. Brian 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.

4. Diane divided the pentagon into three triangles to calculate the measure of x.






Support student problem solving Try not to focus on numerical procedures for deriving answers. Instead, ask students to explain their interpretations and use of the different methods. Raise questions about their assumptions and prompt for explanations based in angle theorems to encourage precision in students’ reasoning. Refer them to the Geometrical Definitions and Properties sheet as needed.

Collaborative analysis of Sample Responses to Discuss (25 minutes) As students complete their solutions, give each group a copy of each of the four Sample Responses to Discuss. You could also display Slide P-3 Instructions for Sample Responses to Discuss.

Four students in another class used Annabel’s, Carlos’s, Brian’s 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 whether the reasoning is correct and complete. Correct the method when necessary. Use the method to calculate the measure of the missing angle x, giving detailed reasons for all your answers.

During small group work, note student difficulties and support student problem solving as before. In particular, think about what students are finding most difficult and use this to focus the next activity; a whole-class discussion.

Whole-class discussion: comparing solution methods (15 minutes) Organize a whole-class discussion comparing the sample solutions methods. Display Slides P-4 to P-7 showing the Sample Responses to Discuss to help with this discussion.

Using your understanding of your students’ difficulties from the assessment task and their work during the lesson, choose one of the sample responses to discuss. Ask one group to present their analysis of that response. Ask for comments and reactions from other students.

[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 that in another way?

Then look at another solution method.

Finally, compare methods.

Which student’s work provided the most complete reasoning?

Which student’s work was most difficult to understand?

The intention is, that students will begin to realize the power of using different methods to solve the same problem and to appreciate the need for, and nature of, adequate reasons for each assertion.

Follow-up lesson: improve individual responses to the assessment task (10 minutes) Return students’ work on Four Pentagons along with a fresh copy of the assessment task sheet.

If you did not write questions on students’ solutions, display them on the board.

Ask students to read through their responses, bearing in mind what they have learned during this lesson.


Look at your original response and think about what you have learned this lesson. Answer the questions [on your scripts/on the board] and revise your response.

Some teachers give this as a homework task.


SOLUTIONS We give examples of some approaches taken by students in trials. There are other methods that lead to correctly reasoned solutions.

Assessment task: Four Pentagons 1. The measure of angle AEJ is 144°.

Explanation 1:

The sum of the measures of the interior angles of an n-gon is 180°(n − 2). For a pentagon this is 180° × 3 = 540°. The pentagons are regular so all their interior angles are congruent. Each interior angle of a regular pentagon is 540° ÷ 5 = 108°. The sum of the angles forming a straight line is 180°. Each exterior angle of a regular pentagon is 180° − 108° = 72°. Angle AEJ is twice the exterior angle of the pentagon = 2 × 72° = 144°.

Explanation 2:

Angle AEF is an exterior angle of a regular pentagon, as is angle FEJ. The sum of the exterior angles of a polygon is 360°. There are five congruent exterior angles in a regular pentagon, each of measure 360° ÷ 5 = 72°. So AEJ = AEF + FEJ = 72° + 72° = 144°.

2. The measure of angle EJF is 36°.

Explanation 1:

Since consecutive angles of a parallelogram are supplementary, angle EJF = 180° − 144° = 36°.

Explanation 2:

The sum of the interior angles of the quadrilateral AFJE is 360°. Since the four pentagons are regular and congruent, sides AF, AE, EJ, JF are equal in length. So AEJF is a rhombus. Opposite angles in a rhombus are congruent. From part 1, angle AEJ = 144° = AFJ. The sum of the angles in a quadrilateral is 360°. 360° − 2 × 144° = 72°. FJE = FAE =

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12

× 72° = 36°.

3. The measure of angle KJM is 108°.

Explanation:

The sum of the measures of the four angles around the point J is 360°. The measure of each of the interior angles in a regular pentagon is 108° and angle EJF is 36° from Question 2. Angle KJM = 360° − (36° + 2 × 108°) = 108°.


From this diagram, we can see that regular pentagons and rhombuses together form a semi-regular tessellation that can be used, for example, as a floor or wall tiling:

Lesson task: The Pentagon Problem

Each method gives a way of calculating the measure of angle x, 75°. Each method uses different definitions and angle properties in the explanation.

1. Annabel’s method In trials some students did not understand the need to justify the assumption that the line “down the middle of the pentagon” bisects the 130° 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 =

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1302

= 65°.

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 CD 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°.

A B c

D

E

F


2. Carlos’s method In trials, some students made 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° –

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12

× 210° = 180° – 105° = 75°.

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. 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°.

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 in trials, perhaps relying on the appearance of the diagram, assumed the three triangles were 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.

Analysis of Sample Responses to Discuss Erasmus used Annabel’s method Erasmus does not justify the claim that the perpendicular bisector of the horizontal side divides the 130° into two equal parts. He could do this by showing that the pentagon is symmetrical so that the bisector of the vertical side passes through the opposite vertex.

He also needs to explain that the perpendicular bisector then divides the pentagon into two congruent quadrilaterals. Then he can apply the property that the sum of the angles in a quadrilateral sum to 360°.

His calculation method is correct but he did not finish his working out.

Erasmus’s use of Annabel’s method gives the correct measure of x = 75°.

Tomas used Carlos’s method Tomas makes a false assumption that all the exterior angles are congruent.

He did not notice that the pentagon is not regular. The exterior angles are all congruent only when the polygon is regular.

Tomas should calculate the size of the exterior angles for each of the known 130° interior angles first.

The angles on a line sum to 180°, so there are three exterior angles of 50°.

360° – 3 × 50° = 360° − 150° = 210°.

So, the two missing exterior angles are congruent and sum to 210°. Each is 210° ÷ 2 = 105°.

Then, since the angles on a line sum to 180°, x + 105° = 180°. So x = 75°.

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Katerina used Brian’s method Katerina is correct that a trapezoid and triangle are formed by the horizontal line, but she does not fully explain her reasoning. It is not clear that the quadrilateral is a trapezoid, or that the trapezoid is isosceles.

She needs to show the base of the quadrilateral is parallel to the top to show that the quadrilateral is a trapezoid.

The horizontal side has at each end the same angle. The slant sides are the same length so the line joining the ends of those slant sides is parallel to the top (trapezoid). The trapezoid is isosceles because the slant sides are equal in length and joined to the top by congruent angles (symmetry) so both base angles can be labeled b.

She is correct that the triangle is isosceles because it has two congruent sides, so the two unknown angles in the triangle are congruent and can both be labeled a.

Katerina made a numerical error in stating a = 50°.

The angles in a triangle sum to 180°.

2a = 180° – 130° = 50°

She had forgotten to divide by two.

Katerina’s next piece of reasoning is faulty.

It is not true that the consecutive angles in every quadrilateral sum to 180°. For example, it is not true that any two consecutive angles in a trapezoid always sum to 180°.

In a trapezoid, the angles formed by a transversal crossing the parallel sides forms a pair of supplementary angles.

Supplementary angles sum to 180°.

So b = 180° – 130° = 50°

Katerina also needs to finish her solution by finding

x = a + b = 25° + 50° = 75°.

Megan used Diane’s method Diane divided the pentagon into three triangles to calculate the measure of x. There is not enough detail to specify a method.

Megan uses faulty reasoning with Diane’s trisection. She makes a false assumption that the triangles are all isosceles.

Megan would need to give reasons to support the assertion that the triangles are isosceles and there are none beyond surface appearance, since they are not!

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b°


Diane’s trisection method can lead to a correct solution. The sum of the angles in a triangle is 180°. So the total angle sum of the pentagon is 3 × 180° = 540°.

This could be provided using the formula for the sum of the angles in a polygon with n sides, 180° (n − 2).

The interior angles sum is 540° and there are three known angles of 130°.

So 2x = 540° – 3 × 130° and x = 75°.

Assuming that the triangles are isosceles leads to a contradiction, showing that the assumption is false. (Proof by contradiction.)

Megan assumes the three triangles formed are all isosceles triangles with two congruent base angles of 65°. Suppose she is correct.

Each has base line of equal length, the base angles of equal measure, two sides of equal length, the apex angles must also be congruent to each other and the triangles are thus congruent.

Each apex angle would be 130° ÷ 3 =

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43 13°.

Since the triangles are isosceles and the angles in a triangle sum to 180°, the two base angles are

(180° −

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43 13

) ÷ 2 =

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68 13°.

x cannot be both

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68 13° and 65°. The assumption leads to a contradiction and must be false.

In Q4, it is not expected that students will show that Megan’s assumption is false. However, we supply a solution in case you want to work on this with students.


Four Pentagons

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.

2. Find the measure of angle EJF.

Explain your reasons and show how you figured it out.

3. Find the measure of angle KJM.

Explain how you figured it out.


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.


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 for answering this problem.

Use each student’s method to calculate the measure of angle x.

Write all your reasoning in detail.











Sample Responses to Discuss

Four students in another class answered The Pentagon Problem using Annabel’s, Carlos’s, Brian’s and Diane’s methods.

Their solutions are shown below.

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.

Make sure to write down all your reasoning in detail.


Erasmus

Erasmus used Annabel’s method


Tomas


Katerina


Megan


Geometrical Definitions and Properties

Isosceles Triangle

An isosceles triangle has at least

two congruent angles and at least

two congruent sides.

Trapezoid

A trapezoid is a

quadrilateral with

at least one pair of

parallel sides.

Regular Polygon

All interior angles of a regular

polygon are congruent. Sides are all

congruent.

Isosceles Trapezoid

An isosceles trapezoid has two

pairs of congruent angles. The slant

sides are congruent.

Angles on a Straight Line

Angles forming a straight line sum

to 180°.

Corresponding Angles

Corresponding angles formed by a

transversal crossing a pair of

parallel lines are congruent.

Supplementary Angles

Supplementary angles formed by

parallel lines crossed by a

transversal are congruent.

Vertical Angles

Vertical angles are congruent.

Angles Around a Point

Angles around a point sum to 360°.

Alternate Interior Angles

Alternate interior angles formed

when parallel lines crossed by a

transversal are congruent.

Exterior Angles of a Polygon

The sum of the exterior angles of a

polygon is 360°.

Interior Angles of a Polygon

The sum of the interior angles of an

n sided polygon is 180 (n-2)°.

Applying Angle Theorems Projector Resources

Instructions for The Pentagon Problem (1)

“This pentagon has three equal sides at the top and two equal sides at the bottom.


Figure out the measure of the angles marked x and explain your reasoning.”

P-1

Diagram is not drawn accurately.

Mrs. Morgan wrote this problem on the board.


Instructions for The Pentagon Problem (2)

•  Four students in Mrs. Morgan’s class came up with different methods for answering 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.

•  Use the Geometrical Definitions and Properties sheet to help.

P-2


Instructions for Sample Responses to Discuss

P-3

Four students answered The Pentagon Problem using Annabel’s, Carlos’s, Brian’s and Diane’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.

•  Make sure to write down all your reasoning in detail.

•  Use the Geometrical Definitions and Properties sheet to help.


Erasmus used Annabel’s method

P-4


Tomas used Carlos’s method

P-5


Katerina used Brian’s method

P-6


Megan used Diane’s method

P-7


A semi-regular tessellation of pentagons and rhombuses

P-8

© 2015 MARS, Shell Center, University of Nottingham This material may be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at http://creativecommons.org/licenses/by-nc-nd/3.0/

All other rights reserved. Please contact [email protected] if this license does not meet your needs.

Mathematics Assessment Project

Classroom Challenges

These materials were designed and developed by the Shell Center Team at the Center for Research in Mathematical Education

University of Nottingham, England:

Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert

with Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead

We are grateful to the many teachers and students, in the UK and the US, who took part in the classroom trials that played a critical role in developing these materials

The classroom observation teams in the US were led by David Foster, Mary Bouck, and Diane Schaefer

This project was conceived and directed for The Mathematics Assessment Resource Service (MARS) by

Alan Schoenfeld at the University of California, Berkeley, and Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the University of Nottingham

Thanks also to Mat Crosier, Anne Floyde, Michael Galan, Judith Mills, Nick Orchard, and Alvaro Villanueva who contributed to the design and production of these materials

This development would not have been possible without the support of Bill & Melinda Gates Foundation

We are particularly grateful to Carina Wong, Melissa Chabran, and Jamie McKee

The full collection of Mathematics Assessment Project materials is available from

http://map.mathshell.org