Extended Essays in Mathematics
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An extended essay (EE) in mathematics is intended for students who are writing on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself. Essays in this group could belong to one of the following five categories:
These are just some of the many different ways that mathematics can be enjoyable or useful, or, as in many cases, both. The list above is just for guidance, there is no requirement that essays should fit wholly within one of these categories. |
Examples:
| Broad Topics | Focused Topics |
| Prime numbers | Prime numbers in cryptography |
| Fractals | The Hausdorff dimension of fractal sets |
| Continued fractions | Continued fractions in birth–death processes |
| CF Gauss: the mathematician | The proof of the law of quadratic reciprocity |
| Graph theory | Using graph theory to minimize cost |
Students’ research should be guided by analysis of primary and secondary sources.
A primary source for research in mathematics involves:
A secondary source of research refers to a comprehensive review of scholarly work, including books, journal articles or essays in an edited collection.
| Topic | The geometry of navigation |
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| Research question | What was the role of mathematics, and geometry in particular, in navigation when we relied on the stars? Does it still play a part now we have man-made satellites? |
| Approach | Using one of the two geometric representations of the Earth (spherical or ellipsoidal), describe how maps and charts were produced to assist navigators in the past. |
| Topic | Square–triangular numbers and Pell’s equation |
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| Research question | How many square numbers are also triangular numbers, where are they and what other problems lead to Pell’s equation? |
| Approach | A description of square and triangular numbers, and how the locations of numbers that are both are solutions of Pell’s equation. Some other problems, perhaps in number theory and geometry, that lead to the equation could be described, with a brief history of the equation included. |
| Topic | The exponential function and the measurement of age and growth |
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| Research question | How does the exponential function, and its calculus, inform areas of science such as nuclear physics, geology, anthropology or demography? |
| Approach | Use one of the settings where exponential growth applies, perhaps modelling the world’s population, to describe the phenomenon. Show how it is applicable in mathematical models of other real situations. |
| Topic | Approximation of irrational numbers by rational numbers |
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| Research question | How well can π, e, 2‾√2 and other irrationals be approximated by rational numbers? |
| Approach | Use the decimal representation of irrational numbers as a starting point to introduce approximation by rationals. Show how a continued fraction expansion of an irrational can also provide rational approximation, and discuss error bounds and orders of approximation. |
| Topic | Archimedes’ calculation of areas |
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| Research question | What is the legacy of Archimedes’ calculations of circular and parabolic areas in today’s methods of integration? |
| Approach | Describe how Archimedes determined the area of a circle by using inscribed polygons, leading also to his measurement of π. Continue with a description of his method of discovery for calculating the area of a parabola. |
Mathematics Sources
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Gale OneFile Databases Gale OneFile databases have two features to help you find search words as well as topics and sub-topics. Use the "subject guide search" feature in Gale OneFile databases to help you find good search terms.(tutorial video). Use the "topic finder" to help you find good topics and sub-topics (tutorial video) |
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In Context Databases Gale In-Context databases offer topic pages and the Topic Finder search feature. This video will show you how to use topic pages.
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National Science Digital Library The NSDL is an open access library of digital content relevant to all aspects of hard and applied sciences. |
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Throughout the EE students should communicate mathematically:
There must be sufficient explanation and commentary throughout the essay to ensure that the reader does not lose sight of its purpose in a mass of mathematical symbols, formulas and analysis.
