Circular Motion
Abstract help
The idea of adding many small quantities in order to gain a good approximation of a desired result is not a new one. Archimedes and others used what is referred to as the "method of exhaustion" to derive values for such measurements as the area of a circle. In fact, inscribed and circumscribed polygons were used to determine this result; small quantities being observed were the central triangles formed by the polygons and the center of the circle, and the increasingly smaller disparity between the area of the polygons and that of the circle. Students will understand the use of tools at hand which differ in different eras.
Georg Friedrich Riemann, a 19th century mathematician, helped clarify this idea. His work was varied and creative, but in elementary calculus he will often be referred to when the concept of Riemann sums is used. This idea had a profound influence on the application of the integral to solving practical problems in which the adding of small quantities had previously been only an approximation process. Passing from approximation to exact value is a power given to us by calculus through the Fundamental Theorem. Students will glean an understanding of the complicated process of expressing new ideas and convincing others of their veracity and relevance.
Georg Friedrich Riemann, a 19th century mathematician, helped clarify this idea. His work was varied and creative, but in elementary calculus he will often be referred to when the concept of Riemann sums is used. This idea had a profound influence on the application of the integral to solving practical problems in which the adding of small quantities had previously been only an approximation process. Passing from approximation to exact value is a power given to us by calculus through the Fundamental Theorem. Students will glean an understanding of the complicated process of expressing new ideas and convincing others of their veracity and relevance.
National Standards help
Representations:
Create and use representations to organize, record, and communicate mathematical ideas;
Connections:
Understand how mathematical ideas build on one another to produce a coherent whole;
Recognize, use and learn about mathematics in contexts outside of mathematics;
Create and use representations to organize, record, and communicate mathematical ideas;
Connections:
Understand how mathematical ideas build on one another to produce a coherent whole;
Recognize, use and learn about mathematics in contexts outside of mathematics;
Pre-requisite Skills help
Teacher Information help
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Student Activity help
To investigate and answer these questions, text and Internet resources should be used.
1. Using the extensive biographical information (available from the links above), determine other mathematicians involved with the definition of the integral and the solution to problems such as finding the area of the circle. What roles did they play? For what mathematical concepts or investigations are they primarily known?
2. How did politics and religion affect the work of mathematicians such as Cauchy, Monge, and Carnot? Was the work of any of these or other mathematicians either delayed or enhanced by their political and religious views?
3. Based on your readings of the working conditions of some of these mathematicians, what were some factors that gave rise to or prevented women from becoming mathematicians and professors?
4. What other problems involved the adding of many very small quantities to achieve an approximation? Who were some of the mathematicians and scientists who used these early methods, preceding the formal and more refined calculus we now use?
1. Using the extensive biographical information (available from the links above), determine other mathematicians involved with the definition of the integral and the solution to problems such as finding the area of the circle. What roles did they play? For what mathematical concepts or investigations are they primarily known?
2. How did politics and religion affect the work of mathematicians such as Cauchy, Monge, and Carnot? Was the work of any of these or other mathematicians either delayed or enhanced by their political and religious views?
3. Based on your readings of the working conditions of some of these mathematicians, what were some factors that gave rise to or prevented women from becoming mathematicians and professors?
4. What other problems involved the adding of many very small quantities to achieve an approximation? Who were some of the mathematicians and scientists who used these early methods, preceding the formal and more refined calculus we now use?
Assessment help
Intranet quizzes and worksheets along with a project of the student's choice
Enrichment / Alternative Activity help
test ... red links
Technology Requirements/Integration help
computers with Internet access