Inductive reasoning: the form of reasoning that goes from the general idea of things to find the specifics of how it works. One example of using this reasoning is found within science. If I conduct a controlled experiment with vinegar and baking soda, and they behave the same way every time I combine them, then I can draw a conclusion about how all baking soda and vinegar will react when combined.
Deductive reasoning: the form of reasoning that goes from the known specifics, to the general, back down to the specifics that we don't know. A geometric proof is the best example of this form of reasoning. We know that line AB is parallel to line CD, an we need to prove that angle ABC is congruent to angle BCD. To prove this conclusion is true, we need to take a step back and see what we know is true about all transversals of parallel lines. We know that the alternate interior angles of transversals of parallel lines are congruent angles. So that means that angle ABC is congruent to angle BCD.
Recognizing patterns and modeling them with equations: the ability of looking at a series of numbers, determining if there is a pattern and if so, what type, and then writing an equation to make any number fit into the pattern.
Special angle relationships: discovering how angle measures behave in certain situations, and how the angles of certain polygons always behave in the same way.
Points of triangle concurrency: The aspects of a triangle that can lead to a conjecture about whether one triangle is congruent to another.
Compass and straight edge constructions: Methods of using a compass and straight edge to create geometric shapes or replications. No measurements required. As long as you have the shape, a compass, and a straightedge, you can make a nearly perfect construction of nearly anything. It is elegantly simple and my favorite aspect of geometry.
Discovering and proving triangle properties: Learning about how triangle components behave when changed and universals about triangle measures, and then proving them according to what we already know as universals.
Discovering and proving polygon properties: Learning universals about polygons and using triangle conjectures to prove the universals are true.
"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." -J.H. Poincare (1854-1912)
This quote is quite poetic, and really true. Will I ever need to know how to calculate complex vectors? Is it crucial that I understand the way that triangles prove that rectangle diagonals are always congruent and bisect each other? Probably not. Do I know it? Yes. Am I glad that I know it? Yes. Why? Because I am one of the rare kids in American society that really enjoys math. I adore the logical way that it works and the glorious way geometric constructions are formed. It is simplistic and extremely complex all at once. It is the foundation of my dream career, and a wonderful challenge. I totally agree with this quote.
Below is a page from my Composition notebook.
