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Three Comprehension Tests are provided for use with this film in the Supplemental Materials page.

Index to Answer Keys:

This test assumes that students have been given the information in the Helpful Background sections entitled Give them the Bird! — Two Examples of Ideas Moving Across Continents and Time. For version of this test in word processing format suitable to modified and/or distributed to a class click here.

1.   What is the great advantage of place-value numbers? Suggested Response: Place-value numbers are important because they allow for easy expression of large numbers and easy mathematical computations with those numbers.

2.   Describe in your own words the concept of place-value numbers. After giving your description, demonstrate it by describing what the numerals 5 and 8 mean in the number 545,680. Suggested Response: Our place-value, ten based numbering system works as follows. The term numeral refers to an Arabic numeral (1,2,3,4,5,6,7,8,9 or 0) within a larger number. The location of an Arabic numeral within the number indicates the power of the base of 10 to which it relates. Because we use a base of 10, the numeral refers to its value multiplied by the power of ten that corresponds to its position within the number. Thus, in the number 545,680, the Arabic numeral "8" is multiplied by 101. This is 8 X 10 or 80. Therefore 80 is added to the value of the number. The numeral 5 appears twice in the number. The first is in the thousands place, which is three digits to the left of the one's place. It is therefore multiplied by 103 (or 1,000). The second time the numeral 5 appears, it is located in the hundreds of thousands place, five digits to the left of the ones place. This second appearance of the numeral 5 is therefore multiplied by 105 or 100,000 and adds 500,000 to the value of the number. In total, the Arabic numeral "5" adds 505,000 to the value of the number.

3.   Why was it so hard for people to come up with the concept of nothing as a number? When did the concept of the zero come to Europe? Suggested Response: The reason that zero was so hard to conceptualize was that numbers were developed for the purpose of counting things that existed, like farm produce, cattle, or sheep, and to measure the distance of lines. People didn't need zero for this. Zero doesn't come naturally when counting objects that exist. The concept of zero was not known in Europe until about the 11th century C.E. when the Europeans learned it from the Arabs who had learned it from the Indians.

4.   Name the first two civilizations that used both the concept of zero as a number in computation and place-value numbers. Give the approximate dates of the development. Suggested Response: The Hindus (Indians) were using the concept by about 600 C.E. and the Maya hundreds of years before. There is no evidence that they were in communication and, apparently, these two civilizations, on different continents and working in isolation, developed the concept independently.

5.   Name two advanced civilizations in Europe that flourished after the concept of place-value numbers was created in Babylon but which did not know this concept. Suggested Response: Ancient Greece and Ancient Rome.

6.   Describe the movement of the concept of zero and place-value numbers to the Americas after the Mayan civilization declined. Suggested Response: Place-value numbers were used in Ancient Babylon beginning about 2500 years ago. The concept of zero as a unit for mathematical calculation arose in India which used the numerals 1,2,3,4,5,6,7,8,9, and 0 by about 600 C.E. The Indians also had place-value numbers. There was a lot of trade between India and Arabia. From India, these concepts traveled on the trade routes to Arabia and from there to Europe. Place-value numbers and zero as a number for computation were used in Europe beginning in the 11th century, 1000 - 1099 C.E. From Europe, the use of zero and place-value numbers traveled across the Atlantic to America with the European colonists.

7.   Name four advanced characteristics of culture developed by the Maya. Suggested Response: Here are seven: (1) calendars; (2) advanced astronomy; (3) the concept of zero and place-value numbers; (4) large stone buildings; (5) large cities; (6) fine metalwork in gold and silver; (7) hieroglyphics, an early form of writing.

8.   Trace the movement of the insult of giving someone "the finger" from the earliest evidence that it was used through to 20th century United States. Suggested Response: Greece to Rome to Europe to North America. The last step could have been directly from Europe to North America or from Europe to Central or South America and from there to North America.

FOR EXTRA CREDIT IF THE TASTE OF CALCULUS UNIT WAS COVERED IN CLASS: 9.  Explain how Archimedes, the ancient Greek mathematician, found the area of a circle and how that relates to calculus. Draw diagrams if it would help in your explanation. (The answer to this question counts for two points.) Suggested Response: Archimedes knew the equation for the area of a triangle and, since a regular polygon can be divided into several triangles, he could calculate the area of a regular polygon. He made his regular polygons smaller and smaller so that they approached closer and closer to the curve of the circle. While he could never get his polygon to eliminate all of the area between the edge of the polygon and the curve of the circle, he found that the values for the area of his many sided polygon approached π times the square of the radius (π r2). This then is a good approximation for the area of a circle. Calculus permits mathematicians to determine the area under a curved line. It does this by finding the area of polygons with small sides that approach the shape of the curve, just as Archimedes made the polygons inscribed within the circle approach the shape of the circle.


This test is designed to teach as well as evaluate. The ELA curriculum materials for this film are contained in the Discussion Questions in the Learning Guide and in this test. For a printable version of the test suitable to distribute to a class (without suggested answers) click here.

1.   Which character changed the most throughout the course of the story? Suggested Response: Probably the best answer will be that it was Angel who changed the most over the course of the film. The other kids went from poor students to disciplined scholars who mastered the AP Calculus exam. Angel, however, was a gang member and had further to go than the other students. An alternative choice is Mr. Escalante who went from being an engineer in a high tech industry to being a teacher and then to achieving the status of a master teacher.

2.   Angel is an interesting character, constantly getting into trouble and constantly figuring a way to get out of it. What drove this character? Suggested Response: There is no one correct answer. Here is an example of a good answer: Angel was an angry young man. Through the course of the movie he realized that his anger wouldn't get him anywhere and that learning calculus would.

3.   At the beginning of the movie, Angel is wearing a hair net. What did that mean? Suggested Response: That he was part of a gang and that he was alienated from school and society.

4.   Remember the scene in which Mr. Escalante pins Angel against a fence and stops him from joining a gang fight? This scene provides important information concerning the character of Mr. Escalante. What does the scene tell the audience (and Angel) about Mr. Escalante? Suggested Response: The possible meanings include: (1) that Mr. Escalante cared for him; (2) that Mr. Escalante knew what was best for him and that Mr. Escalante was willing to put himself at risk to make sure Angel didn't get into trouble; and (3) that Mr. Escalante was a powerful person. Students might come up with additional meanings that we have not thought about.

5.   By the time Angel brought his grandmother to Mr. Escalante's house at Christmas, how had his character changed? Suggested Response: At that point, Angel realized that he wanted to be in the class. He was willing to humble himself to stay in. Credit should also be given if the student remarks on Angel's creativity and willingness to go to any lengths to get back into the class.

6.   Just before the class goes swimming in the ocean to celebrate the exam being over, what is Angel's clothing like? What does this tell us about how his character has changed through the course of the film? Suggested Response: It is neatly pressed and fashionable, not like the gang clothes he wore at the beginning of the film. This shows that Angel has changed from a gangbanger to someone who is going to try to participate in society.

7.   It was difficult for Angel to decide to join the class when they went swimming to celebrate finishing the exam. The scene begins with Angel fully clothed standing on the shore and the rest of the class in the water. What was the significance of Angel taking off his clothes and jumping into the water to be with his classmates? Suggested Response: This action demonstrates that he was committed enough to the kids in the class to allow them to see him in a vulnerable position; something he would never have done had he still been in a gang.

8.   After the first examination results had been invalidated and Angel had provoked the police, his gang friend challenged him to a fight. Why did Angel walk away from that fight and what did this symbolize? Suggested Response: Walking away from a fight will diminish the status of a gang member. Walking away showed that Angel had rejected the ethic of the gang and that he didn't care what his gangbanger friend thought of him. This action symbolizes Angel's separation from the gang life and that his relationship with the gang was no longer important to him.

9.   In this movie, there is one character who is a foil for Mr. Escalante and another who is a foil for Angel. Who are these characters and which of their traits make them foils for Mr. Escalante and for Angel? Suggested Response: The foil for Mr. Escalante was the head of the math department who doubted that the students could learn calculus and who believed that they had probably cheated. The foil for Angel was his gang friend who didn't grow out of the gang life.

10.   The movie opens with a long shot of the Los Angeles River, which is a concrete ditch for much of its length. What is the symbolism of this shot? How does it relate to the rest of the film? Suggested Response: One way to put this is that the Los Angeles river, which has been turned into a sterile concrete ditch with graffiti, is a symbol of the poverty of the lives in the barrio. Another description is that the river, like the children in the movie, exists in an urban environment. They are all suffering under the pressures of urban life, but surviving.



This test assumes that students have been given the information in the Supplemental Materials for this film entitled Public Policy and Burdens of Proof (Perhaps the ETS Gave the Students a Break). Students will need to have available for question # 5 the handout entitled Facts that the ETS Knew or Could Have Discovered in 1982. For a printable version of this test suitable to distribute to a class (without suggested answers) click here.

1.   We are using the term "the presumption of good conduct" to describe the starting point when we evaluate the conduct of anyone in a civil case or when a company evaluates the conduct of a person to make a determination such as a credit score. In criminal cases, there is a similar presumption called "the presumption of innocence." Why don't we just apply the presumption of innocence to civil cases and determinations by private companies? Suggested Response: The reason that "the presumption of innocence" doesn't always apply in civil cases and private determinations is that these proceedings don't determine whether people are guilty of crimes. There is no innocence or guilt in these proceedings. That is reserved for criminal cases.

2.   The chart below mixes up the burdens of proof (on the right) with some of the different situations in which decisions are made in our society (on the left). Match the type of proceeding with the letter assigned to the burden of proof.

Criminal prosecution                A.   preponderance of the evidence
Civil lawsuit for damages                B.   no accepted burden of proof; action is based on a standard chosen by the individual or company
Decision or action by a
person, a business,
or a non-governmental
C.   beyond a reasonable doubt

Suggested Response: In a criminal prosecution, the burden of proof is C: beyond a reasonable doubt. In a civil lawsuit for damages, the burden of proof is A: a preponderance of the evidence. For a decision by a person, a business, or a non-governmental organization the correct answer is B, there is no accepted burden of proof.

3.   The determination of the proper burden of proof in different situations depends upon the facts of the situation and considerations of public policy. Briefly describe some of the public policy considerations supporting the general burdens of proof for criminal prosecutions and for civil lawsuits. Then discuss some of the available options for burdens of proof for individuals and companies. There are some exceptions and extraordinary situations in which the burdens of proof are modified. In your answer, don't worry about the exceptions and extraordinary situations, just refer to the general rules. Suggested Response: Here are some of the public policy considerations that should be considered. A good answer will touch on most of them. (1) Criminal proceedings: These involve a possible loss of freedom which is a very serious penalty. The government, which is very powerful and has many resources, wants to put an individual in jail. The individual usually has very few resources to fight off the government. Even in democratic countries the government sometimes uses the criminal process to persecute those who challenge its policies. The evil of wrongfully putting someone in prison is so great that the public (through the Constitution) has made a public policy determination that it is worth letting many criminals go free to prevent one innocent person from being convicted. Therefore, the people have insisted that it be very difficult for the government to imprison someone. One of the protections is a high burden of proof in criminal proceedings: the government must show guilt beyond a reasonable doubt. (2) Civil lawsuits for damages: The purpose of these proceedings is to provide compensation if someone commits wrongdoing or is negligent and hurts another person. The stakes are property and money. While these are very important, they are less important than someone's liberty. Thus, if a plaintiff can demonstrate that it is more likely than not that another person's wrongdoing or failure to take due care has caused injury to the plaintiff, the plaintiff can recover damages. Thus we have a lower burden of proof: preponderance of the evidence (more likely true than not true). (3) Private Decisions: With respect to individuals, businesses, and non-governmental entities, the usual rule is that they should be free to do what they want, including the choice of a burden of proof that must be satisfied before they take action. However, there are public policies which intervene to restrict this freedom. These include the public policies that people should obey the law and that people should take due care to avoid injuring others. If they fail to do this they can be prosecuted for crimes or be sued for damages in a civil suit.

4.   What burden of proof do you think the ETS should have used in making a determination of whether or not the students cheated? Give your own opinion among the burdens of proof we have discussed in class and briefly describe why you came to that conclusion. Briefly set out the public policies that support your contention. (The answer to this question counts for two points out of ten.) Suggested Response: Any of the burdens of proof discussed in the materials is a potentially correct answer. The three possibilities are: (1) "a preponderance of the evidence" (more likely true than not true; this is the standard used in most civil cases tried in court); (2) "clear and convincing evidence" (highly probable that the fact is true; this is the standard used for decisions of certain issues in civil cases when, for public policy reasons a party is required to provide proof that is more convincing than proof by a preponderance of the evidence; for example, it is often the standard that must be met before punitive damages are assessed) and (3) "beyond a reasonable doubt" (the standard for criminal cases). The justification for applying each of these standards should be well thought out, show that the student has followed the class discussions, and deal with the following two issues: (a) The ETS had to comply with its common law obligation to take reasonable steps to avoid injuring the students under investigation before making a determination that the students had cheated; and (b) the ETS had the right and the responsibility to other test-takers to uphold the integrity of its test.

An example of a good one paragraph answer is: "I believe that the proper standard for the ETS to use in determining if the students cheated is clear and convincing evidence. Being branded a cheater will hurt a student for much of his or her academic career. In addition, these students worked very hard to learn calculus. It is reasonable that the ETS make a determination of cheating only when it is substantially more likely than not that the student cheated. In addition, the ETS is a large organization with all kinds of experts and computers. The interest of the ETS is to prevent cheating. It should be easy for ETS to find out if students cheated using a computerized analysis as well as by looking carefully at the answers."

5.   Apply the burden of proof that you selected in response to the preceding question to the facts known to the ETS when it disqualified the test scores and offered to have the students take the test again. Use the facts found by Mr. Mathews in his investigation except the interviews in which two of the students admitted to cheating on FRQ #6 and then withdrew their admissions. Discuss: (1) whether the burden of proof was sustained with respect to the unusual rate of similar answers to the multiple choice questions and (2) whether the burden of proof was sustained with respect to FRQ #6. (The answer to this question counts for two points out of ten.) Suggested Response: There is no one correct answer as to the multiple choice questions. A good response will deal with most of the facts set out in the handout Facts that the ETS Knew or Could Have Discovered in 1982 and specifically discuss the explanation that Mr. Escalante drilled and re-drilled his students in a systematic approach to calculus problems and that this could very well explain the striking similarities in the answers to the multiple choice questions. As to Free Response Question #6, the fact that the students made an identical error in simplifying a fraction is a telling point and would probably meet all of the possible standards, including "beyond a reasonable doubt." In the answer to both parts of this question, students should not take into account the fact that the students passed the second test supervised by ETS monitors. The question refers to the time just after the students had taken the first test and before their answers were disqualified.

6.   Do you think that the 12 students with identical incorrect answers to FRQ #6 deserved a chance to take the test a second time? Suggested Response: There is no one right answer to this question, and the important concepts conflict. A good answer will be well reasoned and focus on several of the points raised in the class discussion of this issue. These include: the fact that Mr. Escalante drilled and re-drilled the students on a systematic approach to calculus; the fact that the students passed the test without any help from their incorrect response on FRP # 6; the fact that the calculus program at Garfield High was a good program which had just achieved a major breakthrough and that it would be hurt if action were taken against the students.

7.   If a group of middle class students took an AP Calculus test and passed but cheated on a question that they got wrong, should they be permitted to take the test a second time? Assume that the students had no history of cheating. Suggested Response: There is no single correct answer to this question, but it probably should be the same as the answer to the preceding question. This answer should focus on the points in the answer to the preceding question which apply to the situation posed by this question. Some of the points that could have been mentioned in the last response which don't apply, in addition to the difference in socioeconomic class, are that the students described in this question were not under pressure to prove a point (that Latino students from the barrio could learn calculus) and that their success would not have a broad impact. (Note that this question deals with the proper punishment after a determination that the students had cheated, a fact which has not been conclusively proved. On issues of punishment, "Lady Justice" can lift the blindfold a little and consider many things that she could not consider in the determination of the issue of whether the students cheated.)

8.   What effect, if any, does the information that Mr. Mathews developed about whether or not the students cheated have on the core messages of the film? Suggested Response: Reasonable minds could differ. TWM contends that it has no effect. The core messages of the film are that inspired students can achieve wonders and that something very good for math education, for the Latino community, and for the U.S. happened at Garfield High while Mr. Escalante taught there. Whether or not Mr. Escalante's students cheated on one question because of the pressure they were under and their inability to resist temptation, they still learned calculus. It is obvious that they worked hard and mastered the material, given the fact that they passed the examination again when ETS monitors were present. Thus, the evidence that they cheated on FRQ #6 (assuming that they did) may tarnish their reputations but does not negate their achievement in mastering calculus.

Last updated July 19, 2015

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