Computing a correct answer is sufficient for most grade school work, but proving that something must be true is one of the most important skills our students can learn. This ability has immediate benefits for such important tests as USAMTS or the USAMO as well as teaching the skills of reasoning and communication that can help in any profession.
We are very fortunate to once again have a lecture by Dr. Branislav Kisačanin. Here is his description of the upcoming session:
Introduction to Mathematical Proofs
In this lecture we will see examples of common proof techniques: direct proofs, proofs by induction, and proofs by contradiction. The students should have their pencils ready, as they will get a hands-on experience solving and proving mathematical problems.
- As the first illustration of direct proofs we will show that angle bisectors of any triangle have a common intersection point
- As the first illustration of mathematical induction, we will prove Cassini’s identity for Fibonacci numbers
- Finally, as the first illustration of proofs by contradiction, we will look at Euclid’s proof of infinitude of primes
In addition to a few more problems of historical significance, we will work on relatively simple problems that illustrate various proof techniques. In order to make things more interesting, we will let the students figure out which technique to use for each problem.
