This unit, on the Law of Sines and the Law of Cosines, is presented as a guide for making connections to and transitioning from right triangle geometry and trigonometry for students to be able to explore and solve triangles of any type. I believe it can be effectively used to help students construct a deeper knowledge of how to solve triangles of any type before moving to an exploration of the trigonometric functions and their graphs.
The Law of Sines involves using proportions to solve triangles when given ASA, AAS or SSA of a triangle while the Law of Cosines is a generalization of the Pythagorean theorem and can be used to solve triangles when given SAS or SSS of a triangle.
Usually, in most textbooks, treatments of the Law of Sines and the Law of Cosines are presented at the end of the units-after the trigonometric functions are explored.
When I was studying these topics, I wondered how to solve triangles that are not right triangles and thought that moving on before exploring other types of triangles left a hole in my understanding of the topic of solving triangles. Thus, I believe that introducing the Law of Sines and the Law of Cosines early in the course may help students in their understanding of trigonometry by helping them to gain closure on the solution of triangles before moving to analysis of the trigonometric functions.
In developing this unit, The Geometer's Sketchpad is used to help create a dynamical learning environment in which the teacher and students can construct and manipulate figures rather than rely solely on static figures on the pages of a textbook. I intentionally avoided telling what questions to pose because I believe questions will arise within the lessons. The unit can be revised and supplemented to fit the needs of the teacher and the students.
In reflecting on the development of the unit, I realize that for me, it was difficult to determine the teacher audience for which the unit will be appropriate. This led me to printing the proofs etc. on the pages of the plans so that readers can determine if the plans are appropriate for their students. Two proofs of the Law of Cosines are presented for the reader to evaluate and decide if one or both will be appropriate for their lessons.
By having opportunities to observe, practice, and reflect on technology-supported educational activities in which The Geometer's Sketchpad is used to help create a dynamical learning environment where the teacher and students can construct and manipulate figures rather than rely solely on static figures on the pages of a textbook, I feel assured that technology can help facilitate effective learning environments that are community, learner, knowledge, and assessment centered. I have experienced and reflected upon how the components of such a learning environments are interconnected and come together to support effective learning.
I believe that I am ready to use technology-supported mathematics activities within my practice successfully. Although utilizing these kinds of tasks with students who have a wide range of content understandings and learning styles will likely be challenging, I feel better able continue. By having participated in and reflected upon this pedagogy, I now have a better vision of what effective learning environments can look like and feel like, and am poised to continue to enact these kinds of successful teaching and learning situations in my practice.
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