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.  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  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  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

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.  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.

Background Information: Trigonometry  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.  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. S=Opposite Adjacent 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:  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) B) C) 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. 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.

Speed Activity 1.Each student will calculate their speeds in different races: running, skipping, walking, hopping over a measured distance in the schoolyard. 2.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. 3.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. 4.This will repeat for each category: running, skipping, walking, and hopping. 5.The students will then be able to calculate their speed in each category. 6.Students could then determine the top three students in each category, and any other noteworthy placements. 1.Each student will calculate their speeds in different races: running, skipping, walking, hopping over a measured distance in the schoolyard. 2.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. 3.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. 4.This will repeat for each category: running, skipping, walking, and hopping. 5.The students will then be able to calculate their speed in each category. 6.Students could then determine the top three students in each category, and any other noteworthy placements.

Concep Question #2 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 What can be said about this speed graph?

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 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

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 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  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 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.  How does adjusting the angle of a ramp affect an object’s speed?

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.  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.