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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