During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle congruence criteria ASA, SAS, and SSS are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures. Similarity transformations rigid motions followed by dilations define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of "same shape" and "scale factor" developed in the middle grades.
These transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent. The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and, with the Pythagorean Theorem, are fundamental in many real-world and theoretical situations. The Pythagorean Theorem is generalized to non-right triangles by the Law of Cosines.
Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence criterion.
Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Just as the number line associates numbers with locations in one dimension, a pair of perpendicular axes associates pairs of numbers with locations in two dimensions.
This correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa.
The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. In this regard, I'm modestly optimistic about what I've seen in Common Core mathematics, and that it's something of a re-alignment to solid, traditional principles.
However: It's hotly debated, not implemented in all states, not yet fully implemented in the states that have signed on in the last few years, possibly not rigorously tested, not yet representative of what graduating high school students would know, etc. To the extent that schools drifted in a bunch of different directions, it would at best take time for Common Core to wrangle them back on a sensible path.
I will say this: Within the last year I've been giving a 1st-day introduction to all my courses including bullet points such as "This class will include proofs". As part of that I've been asking, "In what high school course did you talk about proofs a lot? So for me that's actually been a bright spot recently, and a point from which I can start making connections and deeper understanding. I teach physics and a little calculus at a community college in California.
My own kids went to local public high school, which is unusually good, and took the IB curriculum. Sometimes in my physics classes, when discussing a topic like Newton's laws or DC circuits, I will use proof-based high school geometry as a reference point for the kind of formal reasoning and rigorous logic that is required.
When I ask for a show of hands, about half my students say that they did two-column proofs in high school geometry. This is usually accompanied with groans. My kids used a high school geometry textbook that did include a heavy dose of theorem proving, although proof was not emphasized as much as it would be if one was taking one's agenda directly from Euclid.
To my taste, the approach was a little ugly and baroque, but the beautiful simplicity of Euclid was definitely embedded somewhere in there. The real number system was treated as something separate from but connected to geometry -- a strange and ugly approach, IMO. I don't know if you're familiar with the US system. Our schools and curricula have traditionally been under completely local control, but over the last 40 years or so, control has gradually been shifting to the states and the federal government.
We have something called Common Core, which is a set of national standards, which showed up around the same time that a well intended but sometimes farcical national political initiative called No Child Left Behind began to collapse under the weight of its impossible promises. Depressingly, Common Core says :.
During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms. If you try to parse this closely, it really doesn't make much sense. How does one do "careful proofs" without starting from "a small set of axioms?
I can share with you my experience. When common-core started to take off, I had just started Geometry. We were expected to prove things regularly, triangle congruency, similarity, angle-sum, parallelograms, Pythagorean theorem, etc.
And we were graded on our reasoning. Here's a list of what I remember we went through, proofs found there way in every part except transformations and constructions though we were expected to still explain what we were doing and why:. This is really a comment on some of the other answers but was too long to add as a comment to other answers, specifically about "short" axioms systems.
I believe that "proofs" tend to be given too much attention in lower grades but let me say something about "axiomatics. The first version of Hilbert's axioms were not independent but in the often used independent version there are 20 axioms, which is not so small a list. The bulk of their time is spent learning to construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.
Students are also expected to know how to find arc lengths and areas of sectors in circles. Expressing geometric properties with equations primarily focuses on two skills. The first is translating between geometric descriptions and equations for conic sections. In simpler terms, this means learning to derive equations for circles, parabolas, ellipses and hyperbolas from given information.
The second skill develops students' abilities to prove simple geometric theorems algebraically. Here the study of geometry asks students to apply their learning from other mathematical domains to the geometric problems. Students studying geometric measurement and dimension take their ability to understand and use formulas a step further as they are called upon to explain the volume formulas for the circumference and area of a circle as well as for the volume of a cylinder, pyramid, cone, sphere and other three dimensional objects.
Students also expand their abilities to visualize relationships between two-dimensional and three-dimensional objects. The final topic in the high school geometry domain is modeling. Modeling makes geometry tangible by bringing it off the paper and into the real world.
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