Joy Bryson

1
Using Trigonometry to Investigate Physical
Concepts Middle School Math and Science

• Joy Bryson

2
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.

3
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.

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Background Information Physics

• 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
  equationMeanwhile, 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

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Background Information Math

• 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 90-degrees to each other.

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Background InformationTrigonometry

• 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 tangent of an
  angle to the ratio of the length of the side
  opposite the angle to the length of the side
  adjacent to the angle.

SOpposite Adjacent
CAdjacent hypotenuse
TOpposite Adjacent
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Teaching Procedures
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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.

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Concep Question 1
B)

• The pythagorean theorem can be used to find the
  missing side for

• A)

C)
D)
E) - all of the above
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• 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.

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Speed Activity

• 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.

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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

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• 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

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Activity

• 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 objects
  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
  objects 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
  objects trip

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Concep Question 3

• It doesnt. Other things affect its speed.
• The more steep the ramp is the faster the object
  travels.
• The more steep the ramp is the slower the object
  travels.

• How does adjusting the angle of a ramp affect an
  objects speed?

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Assessment

• 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 self-evaluation 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.

• 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.