Transcript Document

Joy Bryson
Overview
 This lesson will address the
trigonometry concepts of the
Pythagorean Theorem,and the
functions of sine cosine and tangent.
Those concepts will be used to
investigate the physical ideas of speed
as it relates to inclines and vector
components.
Objectives
 Students will use the Pythagorean
theorem, and the sine, cosine, and
tangent functions to solve for unknown
variables.
 Students will investigate relationships
between angles, speed, and height.
Background Information:
Physics
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Speed refers to "how fast an object is moving." A fast-moving object has a high speed while a
slow-moving object has a low speed. An object with no movement at all has a zero speed.
As an object moves, it often undergoes changes in speed. For example, during an average
trip to school, there are many changes in speed. Rather than the speedometer maintaining a
steady reading, the needle constantly moves up and down to reflect the stopping and starting
and the accelerating and decelerating. At one instant, the car may be moving at 50 mi/hr and
at another instant, it may be stopped (i.e., 0 mi/hr). Yet during the course of the trip to
school the person might average a speed of 25 mi/hr.The average speed during the course of
a motion is often computed using the following equation:Meanwhile, the average velocity is
often computed using the equation:
A website that gives an animated display of these concepts is
http://www.physicsclassroom.com/mmedia/kinema/trip.html
Background Information:
Math
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The Pythagorean theorem is a
mathematical equation which
relates the length of the sides
of a right triangle to the length
of the hypotenuse of a right
triangle.
The Pythagorean theorem is a
useful method for determining
the result of adding two (and
only two) vectors which make a
right angle to each other.
Note: This theorem is not
applicable for adding more than
two vectors or for adding
vectors which are not at 90degrees to each other.
Background Information:
Trigonometry
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Most students recall the meaning of
the useful mnemonic - SOH CAH
TOA which helps students
remember the meaning of the three
common trigonometric functions sine, cosine, and tangent functions.
These three functions relate the
angle of a right triangle to the ratio
of the lengths of two of the sides of
a right triangle.
The sine function relates the sine of
an angle to the ratio of the length of
the side opposite the angle to the
length of the hypotenuse.
The cosine function relates the
cosine of an angle to the ratio of the
length of the side adjacent the
angle to the length of the
hypotenuse.
The tangent function relates the
S=Opposite
tangent of an angle to the ratio of
Adjacent
the length of the side opposite the
angle to the length of the side
adjacent to the angle.
C=Adjacent
hypotenuse
T=Opposite
Adjacent
Teaching
Procedures
Students will learn the concepts of the Pythagorean theorem, sine,
cosine and tangent functions as mentioned in the background
information.
They will focus on the benefit of the trigonometry concepts which is
that they can be used to solve for unknown sides or angles. Many
opportunities will be given to practice the math through physics. For
example:
Question:
 A hiker leaves camp and hikes
11 km, north and then hikes 11
km east. Determine the
resulting displacement of the
hiker.
Question:
 Determine the direction of the
hiker's displacement.
Concep Question #1
 The pythagorean
theorem can be
used to find the
missing side for
A)
C)
B)
D)
E) - all of the above
Students will learn the formula for speed, noting that
in order to determine a speed, you must have a
distance and a time recorded
Students will experiment and calculate average speeds
through various activities and problems.
Quic kTime™ and a
TIFF (Unc ompres s ed) dec ompress or
are needed to s ee this pic ture.
1.
2.
3.
4.
5.
6.
Speed Activity
QuickT ime™ and a
T IFF (Uncompressed) decompressor
are needed to see thi s pi cture.
Each student will calculate their speeds in different races: running, skipping, walking, hopping over a
measured distance in the schoolyard.
Each student will be in a group of three or four, and one person from each group will race against
other people from other group.
While one student from a group is racing against his or her peers, the other group members will time
that student, each one having their own stopwatch. Once that student is done racing, he will have
two or three records of his time and can then calculate the average time he took to run that race.
This will repeat for each category: running, skipping, walking, and hopping.
The students will then be able to calculate their speed in each category.
Students could then determine the top three students in each category, and any other noteworthy
placements.
Concep Question #2
What can be said about this speed graph?
1)
A is faster than B
2)
B is faster than A
3)
A and B have the same
speeds
4)
You cannot tell which one
is the fastest
The two main ideas of trigonometry and physics will be combined into a
lab that requires the students to experiment with angles and lengths
to determine how angles and lengths affect the speed of an object.
Materials will vary, but for all groups will include:
 a Sonic ranger
 A ramp
Qui ckTime™ and a
TIFF ( Uncompressed) decompressor
are needed to see this pi cture.
Activity
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Students will be put into groups of three for the physics lab.
The Problem: Determine if or how inclines may affect the speed of a moving object.
Hypothesis: Students should make a prediction as to what they think will happens to an
object’s speed when the incline gets raised or lowered.
Procedures:
Each group will determine the experiment they want to do. Each group must compare at
least five different angles and the result of each change.
Students will set up at least five different ramps along right angles. The change in angle
would come from changing the heights of the ramp. Using this information and the
trigonometry concepts covered, students will calculate and diagram the angle measures
they used in their varying ramps.
The distances and times that are derived the object’s trip along the ramp, measured by
the sonic ranger device will help in calculation the speeds.
Students would then compare the speeds attained by changing the angle and determine if
there are any conclusions they can draw.
Students would graph the data they collected and present their finding to the class.
Extension Ideas:
 Vary the mass of the object on a constant incline
 Vary the size of the object
 Collect data using a stopwatch instead of the sonic ranger
 Measure the speeds along different points in the object’s trip
Concep Question #3
 How does adjusting
the angle of a ramp
affect an object’s
speed?
1) It doesn’t. Other
things affect its speed.
2) The more steep the
ramp is the faster the
object travels.
3) The more steep the
ramp is the slower the
object travels.
Assessment
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Students will be assessed
based on the conclusions they
are able to draw in their lab,
and from concep questions
Students will also be evaluated
based on in-class observations
made by the teacher.
Students will also do a selfevaluation to evaluate where
they think they stand in their
conceptual understanding.
Students would also be
assessed based on the
individual work done on
class/homework and on
quizzes.
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RUBRIC
4- drew accurate physical
observations and accurately
computed angle measurements.
3- was able to observe the physical
behaviors and drew logical
conclusions, some confusion may
still exist. Was able to accurately
compute the angle measurements.
2- Made errors in observations
and/or failed to draw logical
conclusions about the physical
behaviors. Make errors while
computing the angle
measurements.
1- Made inaccurate observations
and did not make conclusions about
the physical behaviors. Made errors
in the angle measurements.