In 1982, a particular SAT question became famous for being answered incorrectly by every student who attempted it. The question involved two circles, A and B, where the radius of circle A was one-third that of circle B. The problem asked how many revolutions circle A would make while rolling around circle B before its center returned to the starting point. The options provided were A) 3/2, B) 3, C) 6, D) 9/2, or E) 9.
SAT questions are designed to be solved quickly, with students given 30 minutes to tackle 25 problems—roughly one minute per question. Many students, upon seeing this particular problem, instinctively chose option B (three revolutions), reasoning that since the circumference of circle B is three times that of circle A, it would take three full rotations for circle A to roll around circle B. However, this answer, along with all other options, was incorrect.
The reason for the widespread error was that the test writers themselves miscalculated the problem. They believed the answer was three, leading to a situation where no correct answer was listed. This was particularly concerning given the SAT’s long-standing reputation as a critical determinant of students’ futures. A newspaper from that era even stated, “If you mess up on your SAT tests, you can forget it. Your life as a productive citizen is over.”
Out of 300,000 test takers, only three students—Shivan Kartha, Bruce Taub, and Doug Jungreis—took the initiative to write to the College Board, the organization that administers the SAT, to point out the error. Their letters were confident and direct, asserting that the question was wrong and providing proof.
To understand why the answer is not three, we can look at a simpler analogy involving two identical coins. When one coin rolls around another, it rotates not once, but twice, despite both coins being the same size. This phenomenon is known as the coin rotation paradox.
In the case of circles A and B, when circle A rolls around circle B, it completes four rotations in total. This can be demonstrated by creating a scale model of the problem, where one can visually see the rotations as circle A rolls around circle B.
The correct answer to the original SAT question is four revolutions. This can be understood by recognizing that when circle A rolls around circle B, it not only rotates due to its own circumference but also makes an additional rotation because of the circular path it travels.
Interestingly, the wording of the question introduces ambiguity. The term “revolutions” can be interpreted in different ways. In astronomy, a revolution refers to a complete orbit around another body, which would imply that circle A revolves around circle B once. However, if we consider the rotations about its own axis, circle A rotates four times in total.
After reviewing the students’ letters, the College Board publicly acknowledged their mistake and nullified the question for all test takers. This decision led to a rescaling of scores, which could affect students’ final results by up to 10 points out of 800. While this may seem minor, it could impact admissions to universities and scholarships that have strict cutoff scores.
The College Board estimated that rescoring would cost them over $100,000, a financial burden that ultimately fell on test takers. This incident was not the last error on the SAT, but it highlighted the growing concerns about standardized testing, especially as many colleges began to move away from requiring such tests in the wake of the COVID-19 pandemic.
The 1982 SAT question serves as a reminder of the complexities involved in mathematical reasoning and the importance of clarity in problem wording. It also illustrates how errors in standardized testing can have significant repercussions for students. As the landscape of college admissions continues to evolve, the relevance of standardized tests like the SAT is increasingly being called into question.
Using two identical coins, explore the coin rotation paradox. Place one coin on a flat surface and roll the other coin around it. Count the number of rotations the rolling coin makes. Discuss why the rolling coin makes two rotations instead of one, and relate this to the problem of circle A and circle B.
Create a scale model using paper circles to represent circles A and B. Ensure that the radius of circle A is one-third that of circle B. Roll circle A around circle B and count the number of rotations it makes. Discuss how this visual representation helps in understanding the problem.
Read the original SAT question and discuss the ambiguity in the term “revolutions.” Consider how different interpretations could lead to different answers. Write a revised version of the question that eliminates ambiguity and clearly defines what is meant by “revolutions.”
Research the impact of standardized testing errors on students’ futures. Discuss how a mistake like the 1982 SAT question could affect college admissions and scholarships. Consider the financial implications of rescoring tests, such as the estimated cost of $100,000 mentioned in the article.
Hold a class debate on the relevance of standardized tests like the SAT in today’s college admissions process. Consider the shift away from standardized testing in the wake of the COVID-19 pandemic and discuss alternative methods of evaluating student potential.
Circles – A circle is a two-dimensional shape consisting of all points in a plane that are at a given distance from a fixed point, called the center. – In mathematics, the equation of a circle in the Cartesian plane with center $(h, k)$ and radius $r$ is given by $(x-h)^2 + (y-k)^2 = r^2$.
Radius – The radius of a circle is the distance from the center of the circle to any point on its circumference. – If the radius of a circle is $5$ cm, then its diameter is $10$ cm, since the diameter is twice the radius.
Revolutions – A revolution refers to a complete turn around a circle or axis, often used to describe rotational motion. – A wheel makes $3$ revolutions per second, which means it rotates $3$ times around its axis every second.
Circumference – The circumference of a circle is the distance around the circle, calculated as $2pi r$, where $r$ is the radius. – The circumference of a circle with a radius of $7$ cm is $2pi times 7 = 14pi$ cm.
Rotations – Rotation refers to the circular movement of an object around a center or axis. – The Earth completes one rotation on its axis approximately every $24$ hours.
Paradox – A paradox is a statement or concept that contradicts itself or defies intuition. – In physics, the twin paradox arises from the theory of relativity, where a twin traveling at high speed ages slower than the twin who remains stationary.
Analogy – An analogy is a comparison between two things, typically for the purpose of explanation or clarification. – In mathematics, an analogy can be drawn between the flow of electricity in a circuit and the flow of water in pipes.
Problem – A problem in mathematics is a question or puzzle that requires a solution, often involving calculations or logical reasoning. – Solving a quadratic equation is a common problem encountered in algebra.
Testing – Testing in mathematics involves verifying the validity of a solution or hypothesis through various methods and calculations. – After deriving a formula, it is important to conduct testing by substituting known values to ensure its accuracy.
Mathematics – Mathematics is the abstract science of number, quantity, and space, either as abstract concepts or as applied to other disciplines such as physics and engineering. – Mathematics provides the foundation for understanding the laws of physics and the behavior of the universe.
