Sced 416 Group Work, Article Critics 4
07.04.2011
Ahmet Kanık, Rıza Şimşek, Agamurad Bayramov, Ahmet Keklik
Two methods of introducing trigonometry
We know that there are mainly 2 methods in teaching trigonometry. The ratio and the
unit circle method. In the article, it was stated that if we are only interested in students` ability
of quickly performing the solution of triangles and and the associated practical applications
then the ratio method is better to be preferred. But as far as the conceptual development of
students is concerned, the unit cicle method of trigonometric ratios should be chosen. It
provides more extension beyond the first quadrant and concrete meanings for the
trigonometric functions, as lengths of intervals which can be directly measured. However, the
problem with the unit circle method is not the unit circle. Rather it is the complex procedure
that are required to use the resulting definitions to solve triangles. To balance this, the
preferred approach should be that basic trigonometry be introduced using the unit circle
trigonometric function definitions (and in more than one quadrant!), connecting them to the
ratio definitions and then adopting the techniques of the ratio method for the solution of
triangles.
Activities for Students: Trigonometry through a Ferris Wheel
Up to now, we have reflected on Thompson’s conceptual understanding ideas related
to trigonometry and then connect his ideas to history of trigonometry and try to see how the
concept developed in history. Last week we have been talking on “reflections on history of
trigonometry” and we agreed on that rather than right triangle trigonometry, today there is
need to teach and learn periodic functions more than before. So this week we takled on an
article about an activity that is all about periodic functions and the effects of constants on the
graph of periodic functions. Generally, students study phase shifts, horizontal and vertical
translations, and changes in period so that they can sketch the graph of generalized functions
such as
by recognizing the information provided by the
constants a, b, c, and d,. Far too often, students master this material by memorizing it and thus
have litt1e comprehension of why and how each value in an equation affects the graph. One
of my favorite projects is designed not only to review and solidify this material but also to
engage students in an experiential learning opportunity that requires precise manipulation of
these constants in a realistic setting” ( Kaplan G. 2008). The effect of these constants usually
is memorized by students as well as memorizing a period of any periodic function that for
example period of sin is 2pi/a if or pi/a according to the power of the function.
“The project involves building a Ferris wheel from K'nex construction materials,
gathering data, and finding a particular trigonometric function to represent the height of a
particular seat on the Ferris wheel during its revolutions. Each group brainstorms to find a
method for determining the height of a small plastic figure placed in one of the seats at any
given time.” ( Kaplan G. 2008). By this activity sudents investigate periodic functions in real
context and play with real data. So they construct the consept of periodict by this logico
mathematical activity. They share ideas and discuss on a more realistic context. Gathering the
data provides an exceptional opportunity for student creativity and problem solving.
Some of the questions asked during the task are: “In paragraph form, describe below the
following process to obtain a function that models the height particular of a seat on the Ferris wheel as
it revolves.
 Make an educated guess of a trigonometric function that best fits these data and explain why
you have chosen this particular function. Describe how the numbers in the function relate to
the diameter of the Ferris wheel and the period of the rotation.
 Examine the graph of your function on your calculator. Describe how to modify one of the
constants so that the function more accurately fits your data.
 Test the modified constant by examining the graph of your function. Describe how to further
modifv one of the constants so that the function more accuratery fits your data.
 Repeat steps (D) and (c) until your conjecture looks "close." For each function, describe how
the numbers in the formula relate to the diameter of the Ferris wheel and the period of the
rotation” ( Kaplan G. 2008).
The questioning pattern above helps students reason the steps and sustain higher level
thinking which brings a more profound conceptual understanding of trigonometric functions.
There is similar lesson plan “Graphs from the Unit Circle” at NTCM’s illuminations web
page. Students are using spagetti to discover some properties of sine and cosine functions
and graph these functions. The materials are easily available rather than designing the above
electronic ferris wheel. So it is better to rethink on whether the activity meet our objectives or
not.
Resources: Kaplan G. Activities for Students: Trigonometry through a Ferris Wheel,
September 2008, Volume 102, Issue 2, Page 138
Stacey K. Two methods of introducing trigonometry, University of Melbourne, Australia