Question Details: Radius of Circle A is 1/3 the radius of Circle B. How many revolutions does Circle A make while rolling around B?
Options: A) 3/2, B) 3, C) 6, D) 9/2, E) 9
Context: 30 minutes to solve 25 problems, creating pressure for quick answers.
Initial Intuition and Mistake
Common Misconception: Answer B (3 revolutions) seems logical because Circle B's circumference is three times that of Circle A.
Reality: All listed options (A, B, C, D, E) are incorrect. Actual answer was not provided in options.
Significance: The SAT has a reputation for accuracy, and such mistakes are typically avoided.
Student Responses and Outcome
Students’ Actions: Only three out of 300,000 reported the error: Shivan Kartha, Bruce Taub, Doug Jungreis.
Students' Confidence: They stated that the test writers were wrong and backed their claims with evidence.
College Board Response: They eventually acknowledged the mistake and nullified the question.
Coin Rotation Paradox Explanation
Understanding the Paradox: When rolling two identical coins, one rotates twice instead of once when rolling around the other.
Modeling the Problem: The actual answer to the circle problem is four revolutions due to the added rotation from the circular path.
Explanation Techniques:
Use a scale model to visualize the rotations.
Wrap a ribbon around Circle B to measure the distance.
General Solution to Similar Problems
Methodology: Calculate the ratio of circumferences and add one for the circular path.
Different Perspectives: From Circle B's perspective, Circle A rotates three times when viewed externally; internally, it rotates four times.
Defining Revolutions
Ambiguity in Terminology: "Revolution" can mean different things (e.g., completing a full orbit vs. rotating about one’s own axis).
Astronomical Definition: Circle A revolves around Circle B once, leading to the ambiguity of the question.
Impact of the Error on Students
Score Adjustments: Rescoring affected students' scores by up to 10 points, which can impact admissions to universities.
Financial Implications: Rescoring costs exceeded $100,000 for the testing service.
Broader Discussion on Standardized Testing
Declining Importance: Post-COVID-19, many colleges have stopped requiring standardized tests like the SAT.
Personal Reflection: Discussion on the speaker's own SAT experience and writing of math problems for competitions.
Conclusion and Learning Opportunities
Hands-on Exploration: Emphasized the importance of hands-on learning in understanding complex concepts.
Sponsorship by Brilliant: Encouragement to engage with educational platforms like Brilliant for interactive learning.
Final Thoughts
Understanding Mathematical Concepts: The lecture highlights the significance of understanding concepts through exploration rather than purely theoretical approaches.