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Introduction and Rationale:
This Precalculus lesson begins with the
students viewing the animation of a Ferris Wheel using the software
program Geometer's Sketchpad (See the link below for a video of
the initial project and student creations). Students are given
the following broad directions for the activity.
Directions.doc The activity has students working in groups to
accomplish this task. The students measure and relate quantities with
accuracy that would not be possible without the technology. In
addition, Geometer's Sketchpad allows for the exploration of
ideas since the initial task allows for a great deal of freedom for
investigation.
A primary premise to this approach is the use of situation that is
experientially (or imaginatively) real to motivate the relevant
mathematics. (See:
Realistic Mathematics Education for a description of this Learning
Theory) This particular series of lessons is the beginning of an
entire unit on trigonometry that will encompass most of the 3rd
nine weeks of the course. However, the students are not aware of the
content (Trigonometry) that is related to the motion of the Ferris
wheel. They will not be predisposed to think about trigonometry.
In most textbooks the application problems come only after the content
is learned. This process uses the applications to develop the
mathematics. Overall this produces a much more authentic approach
to problem solving and mathematics. Through exploration (and some
direction/suggestion from the teacher) students "discover" relevant
mathematics packed inside of the real world application of a Ferris
wheel.
Objectives:
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Root
the topics of trigonometry in the discussion and analysis of the
experiential world/mind of the students.
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Create a model that explains the motion of a person riding a Ferris
wheel.
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Introduce the concepts of trigonometry
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Consider angles greater than 90 degrees
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Reason positive and negative values within a circle
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Develop meaningful connections between measured quantities
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Interpret and analyze constructed graphs
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Link
the sine function to the vertical distance (or height from the
ground)
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Link
the cosine function to the horizontal distance from center
Learning Activities/Outline:
This
lesson encompasses 2 days of classroom instruction. (January 10th
and 11th 2008)
The
directions and animation will be provided on day 1. At the end of
the first day, the students will report on the quantities that they are
focusing upon. This provides a specific time for teacher and class
feedback for the groups. At the end of the second day, groups will
give a brief presentation of their findings (graphs, diagrams, and
conclusions). The teacher will provide a summary upon the end of
the second day. Students will be responsible for a written
reflection in Blackboard.
Laptop Implications:
The
activities for this lesson were designed specifically for the laptops.
Without the use of Geometer's Sketchpad, it would be extremely
difficult to complete this project. The computers easily allow the
students to explore multiple concepts with creativity. The focus
on the activity remains on the exploration rather than on measurement of
quantities within a circle. In addition, the students easily could
project their finished products and describe their conclusions of the
activity for the entire class.
Role of the Teacher:
The role
of the teacher is somewhat different than the traditional role
throughout this lesson. The teacher simply introduces the topic
and provides the directions to the class. The goal of the teacher
is NOT to tell them what mathematical relationships to find
through the exploration of the Ferris wheel. Rather, the teacher
is the facilitator of the classroom discussions. The teacher does
provide guidance with ideas to investigate. This is accomplished
specifically during the end of the first day when the students discuss
as a class the quantities that they are planning to investigate.
If their quantities would not lead to productive mathematics, they are
encouraged in a new direction. The teacher is also responsible for
the summary at the end of the two days---making the connections between
the students presentations by looking for common elements, explaining
the reasons for the observed differences, and presenting a unified
viewpoint.
Student Reactions:
The
students "discovered" significant elements of trigonometry. Their
creations and presentations included the trigonometric concepts
of: coterminal angles, sine and cosine waves, cotangent and tangent
functions, asymptotes, right triangles, reference angles, linear
relationship between angle and arc length, and more.
The
students responded very favorably to the project. Many commented
about the improved ability to connect meaning to trigonometric ideas.
Here are some excerpts of their Blackboard reflections and comments.
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I felt the Ferris wheel
project was very helpful in allowing me to better understand
circular motion and how it relates to sine, cosine, and tangent.
When we first started, only a few variables to measure for a circle
came to mind. I never really considered how the Ferris wheel moves
both vertically and horizontally. It was very helpful to use
sketchpad because it provided a great visual that I feel will help
me as we continue to study circular motion.
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This
project has made the motion of the circle more clear to me. Seeing
the graphs and working with Sketchpad made it easier to see what was
going on and to see the relationships between different parts of a
circle. Seeing it visually made me understand how trigonometry
is applied to real life.
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I
was surprised by what the graphs we created looked like, but I
learned how to interpret them and understand why they looked the way
they did.
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When he animated the
Ferris wheel, and the graphs began to move with the motion of the
circle, was very easy to understand and appealing. Analyzing the
graphs and trying to figure out the relationships made learning
about the motion of the circle a lot easier.
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Until now we've just been shown the graphs and told to memorize
them. Now I know where exactly they come from.
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A
moving diagram is a good idea, because sometimes it is hard to
visualize motion in your head, especially when the two components
could be so similar (i.e. vertical height from x-axis, or vertical
height from a fixed point ON the x-axis).
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I am
very happy with the results my class discovered, at first I was
apprehensive since such minimal instructions were given, but now I
understand why there weren't any specific relationships given to us
to test. Mr. Thompson was simply getting us to think and realize why
those relationships made the different types of graphs.
Illinois Learning Standards:
8.B.4a
Represent algebraic concepts with physical materials, words, diagrams,
tables, graphs, equations and inequalities and use appropriate
technology.
8.B.5
Use
functions including exponential, polynomial, rational, parametric,
logarithmic, and trigonometric to describe numerical relationships.
8.C.5 Use
polynomial, exponential, logarithmic and trigonometric functions to
model situations.
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