Angles of Light: Rainbows and 8th grade math
Objective:
We will take a look at the properties of light! How does light travel? Can you explain it in mathematical language that includes angles of reflection and refraction.
Start with refresher on circles:
There is 360-degrees in a circle. What does that mean? One full rotation around a circle puts us back in the same spot we started. What if we go half way around? How many degrees would that be? Where would we be on that circle?
Lesson: The Line of Sight:
Without light, there would be no sight. With our eyes, humans and other animals have “visual ability”. That is the result of the complex interaction of light, eyes and brain. We are able to see because light from an object can move through space and reach our eyes. Once light reaches our eyes, signals are sent to our brain, and our brain deciphers the information in order to detect the appearance, location and movement of the objects we are sighting at. The whole process, as complex as it is, would not be possible if it were not for the presence of light. Without light, there would be no sight.
If you were to turn off the room lights for a moment and then cover all the windows with black construction paper to prevent any entry of light into the room, then you would notice that nothing in the room would be visible. There would be objects present that were capable of being seen. There would be eyes present that would be capable of detecting light from those objects. There would be a brain present that would be capable of deciphering the information sent to it. But there would be no light! The room and everything in it would look black. The appearance of black is merely a sign of the absence of light. When a room full of objects (or a table, a shirt or a sky) looks black, then the objects are not generating nor reflecting light to your eyes. And without light, there would be no sight.
Review of Angles: You can think of an angle as the sides of a wedge, like the cut sides of a slice of pizza. Or, you can think of an angle as a point with two sides extending from the point, like branches on a tree. In all of these examples, since each side extends from a point in only one direction, the sides are called half lines or rays. The point is called the vertex of the angle. Practice: A. Sketch the angles made by these turns. For each sketch, include a curved arrow indicating the turn and label the angle with its degree measure. 1. One third of a right-angle turn 2. Two thirds of a right-angle turn 3. One quarter of a right-angle turn 4. One and a half right-angle turns 5. Two right-angle turns 6. Three right-angle turns B. In parts (1)–(6), sketch an angle with approximately the given measure. For each sketch, include a curved arrow indicating the turn. 1. 20-degrees 2. 70-degrees 3. 150-degrees 4. 180-degrees Information: Mathematicians and scientists find it useful to locate points using different kinds of coordinate grids.
One way to locate points is to use a circular grid. On this kind of grid, angle measures help describe the location of points. Two examples of circular grids are shown below. Number the grids moving out from the center at 0. Place point D at (2, 45-degrees) and point E somewhere on the second grid.
Points on a circular grid are described by giving a distance and an angle. For example, point D has coordinates (2, 45).To locate a point, start at the center of the grid and move to the right the number of units indicated by the first coordinate. Then, move counterclockwise along that circle the number of degrees given by the second coordinate. To locate (2, 45), move to the right 2 units on the degree line and then move up (around the circle) to the 45 line. What are the coordinates of point E above? Use of angle ruler: There are several tools for measuring angles. One of the easiest to use is the angle ruler. An angle ruler has two arms, like the sides of an angle. A rivet joins the arms. This allows them to swing apart to form angles of various sizes. One arm is marked with a circular ruler showing degree measures from 0-degrees to 360-degrees.
To measure an angle with an angle ruler, first place the rivet over the vertex of the angle. Then set the center line of the arm with the ruler markings on one side of the angle. Swing the other arm around counterclockwise until its center line lies on the second side of the angle. The center line on the second arm will pass over a mark on the circular ruler. This tells you the degree measure of the angle. Practice on these angles:
In 1937, the famous aviator Amelia Earhart tried to become the first woman to fly around the world. She began her journey on June 1 from Miami, Florida. She reached Lae, New Guinea, and then headed east toward Howland Island in the Pacific Ocean. She never arrived at Howland Island. In 1992, 55 years later, investigators found evidence that Earhart had crashed on the deserted island of Nikumaroro, far off her intended course. It appears that an error may have been made in plotting Earhart’s course. How many degrees off course was Earhart’s crash site from her intended destination? Find the measure of the angle labeled x, without measuring. Luminous versus Illuminated ObjectsThe objects that we see can be placed into one of two categories: luminous objects and illuminated objects. Luminous objects are objects that generate their own light.Illuminated objects are objects that are capable of reflecting light to our eyes. The sun is an example of a luminous object, while the moon is an illuminated object. During the day, the sun generates sufficient light to illuminate objects on Earth. The blue skies, the white clouds, the green grass, the colored leaves of fall, the neighbor's house, and the car approaching the intersection are all seen as a result of light from the sun (the luminous object) reflecting off the illuminated objects and traveling to our eyes. Without the light from the luminous objects, these illuminated objects would not be seen.
To help with geometric understanding, we will assume that light travels in rays. We begin with light rays moving through the air at a constant speed and consider the reflection of light. In 1657 the mathematician Pierre de Fermat postulated a simple principle: Light follows a path that minimizes total travel time.
Activity: Combined with Connected Math 2 (Two-dimensional geometry)
Pg 24 Polygons and Angles
Exercise 2.1 Understanding Angles:
Materials:
a) Laser pointer or flash light
b) Angle ruler/protractor
c) Mirror
Place the protractor above the mirror and shine the light at different angles toward the mirror. Draw 3 angles at 3 different degrees. Label all parts of the path the light takes. (Include ray, vertex and degrees)
Challenge question: Can you make a 90-degree angle with light?
When viewing the image of the object in a plane mirror, one of these rays of light originates at the object location and first travels along a line towards the mirror (as represented by the blue ray in the diagram below). This ray of light is known as the incident ray - the light ray approaching the mirror. The incident ray intersects the mirror at the same location where your line of sight intersects the mirror. The light ray then reflects off the mirror and travels to your eye (as represented by the red ray in the diagram below); this ray of light is known as the reflected ray.
So the manner in which light travels to your eye as you view the image of an object in a mirror can be summarized as follows.
To view the image of an object in a mirror, you must sight along a line at the image. One of the many rays of light from the object will approach the mirror and reflect along your line of sight to your eye.
The second important idea that can be gleaned from this stoppered pencil lab pertains to the location of the image. Observe in the diagram above that the image is positioned directly across the mirror along a line that runs perpendicular to the mirror. The distance from the mirror to the object (known as the object distance) is equal to the distance from the mirror to the image (known as the image distance). For all plane mirrors, this equality holds true:
Object distance = Image distance
Law of Reflection:
Light is known to behave in a very predictable manner. If a ray of light could be observed approaching and reflecting off of a flat mirror, then the behavior of the light as it reflects would follow a predictable law known as the law of reflection. The diagram below illustrates the law of reflection.
In the diagram, the ray of light approaching the mirror is known as the incident ray(labeled I in the diagram). The ray of light that leaves the mirror is known as the reflected ray (labeled R in the diagram). At the point of incidence where the ray strikes the mirror, a line can be drawn perpendicular to the surface of the mirror. This line is known as a normal line (labeled N in the diagram). The normal line divides the angle between the incident ray and the reflected ray into two equal angles. The angle between the incident ray and the normal is known as the angle of incidence. The angle between the reflected ray and the normal is known as the angle of reflection. (These two angles are labeled with the Greek letter "theta" accompanied by a subscript; read as "theta-i" for angle of incidence and "theta-r" for angle of reflection.) The law of reflection states that when a ray of light reflects off a surface, the angle of incidence is equal to the angle of reflection.
Reflection and the Locating of Images To view an image of a pencil in a mirror, you must sight along a line at the image location. As you sight at the image, light travels to your eye along the path shown in the diagram below. The diagram shows that the light reflects off the mirror in such a manner that the angle of incidence is equal to the angle of reflection.
It just so happens that the light that travels along the line of sight to your eye follows the law of reflection. If you were to sight along a line at a different location than the image location, it would be impossible for a ray of light to come from the object, reflect off the mirror according to the law of reflection, and subsequently travel to your eye. Only when you sight at the image, does light from the object reflect off the mirror in accordance with the law of reflection and travel to your eye. This truth is depicted in the diagram below.
For example, in Diagram A above, the eye is sighting along a line at a position above the actual image location. For light from the object to reflect off the mirror and travel to the eye, the light would have to reflect in such a way that the angle of incidence is less than the angle of reflection. In Diagram B above, the eye is sighting along a line at a position below the actual image location. In this case, for light from the object to reflect off the mirror and travel to the eye, the light would have to reflect in such a way that the angle of incidence is more than the angle of reflection. Neither of these cases would follow the law of reflection. In fact, in each case, the image is not seen when sighting along the indicated line of sight. It is because of the law of reflection that an eye must sight at the image location in order to see the image of an object in a mirror.
Check Your Understanding 1. Consider the diagram at the right. Which one of the angles (A, B, C, or D) is the angle of incidence? ______ Which one of the angles is the angle of reflection? ______
Answer: Angle B is the angle of incidence (angle between the incident ray and the normal). Angle C is the angle of reflection (angle between the reflected ray and the normal).
2. A ray of light is incident towards a plane mirror at an angle of 30-degrees with the mirror surface. What will be the angle of reflection?
Answer: The angle of reflection is 60 degrees. (Note that the angle of incidence is not 30 degrees; it is 60 degrees since the angle of incidence is measured between the incident ray and the normal.)
3. Perhaps you have observed the image of the sun in the windows of distant buildings near the time that the sun is rising or setting. However, the image of the sun is not seen in the windows of distant building during midday. Use the diagram below to explain, drawing appropriate light rays on the diagram.
Answer: The ray of light drawn from the sun’s position at 7pm to the distant window reflects off the window and travels to the observer’s eye. On the other hand, a ray of light drawn from the 1pm sun position to the window will reflect and travel to the ground, never making it to the distant observer’s eye.
4. A ray of light is approaching a set of three mirrors as shown in the diagram. The light ray is approaching the first mirror at an angle of 45-degrees with the mirror surface. Trace the path of the light ray as it bounces off the mirror. Continue tracing the ray until it finally exits from the mirror system. How many times will the ray reflect before it finally exits?
Answer: The light reflects twice before it finally exits the system. Draw a normal at the point of incidence to the first mirror; measure the angle of incidence (45 degrees); then draw a reflected ray at 45 degrees from the normal. Repeat the process for the second mirror.
FUN FACTS: SPEED OF LIGHT in air was measured accurately by Albert Michelson, who won the Nobel prize in 1907 for his work. Using a rotating eight-sided mirror, he measured the time light took to make a round-trip between mountain tops that were a known distance apart.
Objective:
We will take a look at the properties of light! How does light travel? Can you explain it in mathematical language that includes angles of reflection and refraction.
Start with refresher on circles:
There is 360-degrees in a circle. What does that mean? One full rotation around a circle puts us back in the same spot we started. What if we go half way around? How many degrees would that be? Where would we be on that circle?
Lesson: The Line of Sight:
Without light, there would be no sight. With our eyes, humans and other animals have “visual ability”. That is the result of the complex interaction of light, eyes and brain. We are able to see because light from an object can move through space and reach our eyes. Once light reaches our eyes, signals are sent to our brain, and our brain deciphers the information in order to detect the appearance, location and movement of the objects we are sighting at. The whole process, as complex as it is, would not be possible if it were not for the presence of light. Without light, there would be no sight.
If you were to turn off the room lights for a moment and then cover all the windows with black construction paper to prevent any entry of light into the room, then you would notice that nothing in the room would be visible. There would be objects present that were capable of being seen. There would be eyes present that would be capable of detecting light from those objects. There would be a brain present that would be capable of deciphering the information sent to it. But there would be no light! The room and everything in it would look black. The appearance of black is merely a sign of the absence of light. When a room full of objects (or a table, a shirt or a sky) looks black, then the objects are not generating nor reflecting light to your eyes. And without light, there would be no sight.
Review of Angles: You can think of an angle as the sides of a wedge, like the cut sides of a slice of pizza. Or, you can think of an angle as a point with two sides extending from the point, like branches on a tree. In all of these examples, since each side extends from a point in only one direction, the sides are called half lines or rays. The point is called the vertex of the angle. Practice: A. Sketch the angles made by these turns. For each sketch, include a curved arrow indicating the turn and label the angle with its degree measure. 1. One third of a right-angle turn 2. Two thirds of a right-angle turn 3. One quarter of a right-angle turn 4. One and a half right-angle turns 5. Two right-angle turns 6. Three right-angle turns B. In parts (1)–(6), sketch an angle with approximately the given measure. For each sketch, include a curved arrow indicating the turn. 1. 20-degrees 2. 70-degrees 3. 150-degrees 4. 180-degrees Information: Mathematicians and scientists find it useful to locate points using different kinds of coordinate grids.
One way to locate points is to use a circular grid. On this kind of grid, angle measures help describe the location of points. Two examples of circular grids are shown below. Number the grids moving out from the center at 0. Place point D at (2, 45-degrees) and point E somewhere on the second grid.
Points on a circular grid are described by giving a distance and an angle. For example, point D has coordinates (2, 45).To locate a point, start at the center of the grid and move to the right the number of units indicated by the first coordinate. Then, move counterclockwise along that circle the number of degrees given by the second coordinate. To locate (2, 45), move to the right 2 units on the degree line and then move up (around the circle) to the 45 line. What are the coordinates of point E above? Use of angle ruler: There are several tools for measuring angles. One of the easiest to use is the angle ruler. An angle ruler has two arms, like the sides of an angle. A rivet joins the arms. This allows them to swing apart to form angles of various sizes. One arm is marked with a circular ruler showing degree measures from 0-degrees to 360-degrees.
To measure an angle with an angle ruler, first place the rivet over the vertex of the angle. Then set the center line of the arm with the ruler markings on one side of the angle. Swing the other arm around counterclockwise until its center line lies on the second side of the angle. The center line on the second arm will pass over a mark on the circular ruler. This tells you the degree measure of the angle. Practice on these angles:
In 1937, the famous aviator Amelia Earhart tried to become the first woman to fly around the world. She began her journey on June 1 from Miami, Florida. She reached Lae, New Guinea, and then headed east toward Howland Island in the Pacific Ocean. She never arrived at Howland Island. In 1992, 55 years later, investigators found evidence that Earhart had crashed on the deserted island of Nikumaroro, far off her intended course. It appears that an error may have been made in plotting Earhart’s course. How many degrees off course was Earhart’s crash site from her intended destination? Find the measure of the angle labeled x, without measuring. Luminous versus Illuminated ObjectsThe objects that we see can be placed into one of two categories: luminous objects and illuminated objects. Luminous objects are objects that generate their own light.Illuminated objects are objects that are capable of reflecting light to our eyes. The sun is an example of a luminous object, while the moon is an illuminated object. During the day, the sun generates sufficient light to illuminate objects on Earth. The blue skies, the white clouds, the green grass, the colored leaves of fall, the neighbor's house, and the car approaching the intersection are all seen as a result of light from the sun (the luminous object) reflecting off the illuminated objects and traveling to our eyes. Without the light from the luminous objects, these illuminated objects would not be seen.
To help with geometric understanding, we will assume that light travels in rays. We begin with light rays moving through the air at a constant speed and consider the reflection of light. In 1657 the mathematician Pierre de Fermat postulated a simple principle: Light follows a path that minimizes total travel time.
Activity: Combined with Connected Math 2 (Two-dimensional geometry)
Pg 24 Polygons and Angles
Exercise 2.1 Understanding Angles:
Materials:
a) Laser pointer or flash light
b) Angle ruler/protractor
c) Mirror
Place the protractor above the mirror and shine the light at different angles toward the mirror. Draw 3 angles at 3 different degrees. Label all parts of the path the light takes. (Include ray, vertex and degrees)
Challenge question: Can you make a 90-degree angle with light?
When viewing the image of the object in a plane mirror, one of these rays of light originates at the object location and first travels along a line towards the mirror (as represented by the blue ray in the diagram below). This ray of light is known as the incident ray - the light ray approaching the mirror. The incident ray intersects the mirror at the same location where your line of sight intersects the mirror. The light ray then reflects off the mirror and travels to your eye (as represented by the red ray in the diagram below); this ray of light is known as the reflected ray.
So the manner in which light travels to your eye as you view the image of an object in a mirror can be summarized as follows.
To view the image of an object in a mirror, you must sight along a line at the image. One of the many rays of light from the object will approach the mirror and reflect along your line of sight to your eye.
The second important idea that can be gleaned from this stoppered pencil lab pertains to the location of the image. Observe in the diagram above that the image is positioned directly across the mirror along a line that runs perpendicular to the mirror. The distance from the mirror to the object (known as the object distance) is equal to the distance from the mirror to the image (known as the image distance). For all plane mirrors, this equality holds true:
Object distance = Image distance
Law of Reflection:
Light is known to behave in a very predictable manner. If a ray of light could be observed approaching and reflecting off of a flat mirror, then the behavior of the light as it reflects would follow a predictable law known as the law of reflection. The diagram below illustrates the law of reflection.
In the diagram, the ray of light approaching the mirror is known as the incident ray(labeled I in the diagram). The ray of light that leaves the mirror is known as the reflected ray (labeled R in the diagram). At the point of incidence where the ray strikes the mirror, a line can be drawn perpendicular to the surface of the mirror. This line is known as a normal line (labeled N in the diagram). The normal line divides the angle between the incident ray and the reflected ray into two equal angles. The angle between the incident ray and the normal is known as the angle of incidence. The angle between the reflected ray and the normal is known as the angle of reflection. (These two angles are labeled with the Greek letter "theta" accompanied by a subscript; read as "theta-i" for angle of incidence and "theta-r" for angle of reflection.) The law of reflection states that when a ray of light reflects off a surface, the angle of incidence is equal to the angle of reflection.
Reflection and the Locating of Images To view an image of a pencil in a mirror, you must sight along a line at the image location. As you sight at the image, light travels to your eye along the path shown in the diagram below. The diagram shows that the light reflects off the mirror in such a manner that the angle of incidence is equal to the angle of reflection.
It just so happens that the light that travels along the line of sight to your eye follows the law of reflection. If you were to sight along a line at a different location than the image location, it would be impossible for a ray of light to come from the object, reflect off the mirror according to the law of reflection, and subsequently travel to your eye. Only when you sight at the image, does light from the object reflect off the mirror in accordance with the law of reflection and travel to your eye. This truth is depicted in the diagram below.
For example, in Diagram A above, the eye is sighting along a line at a position above the actual image location. For light from the object to reflect off the mirror and travel to the eye, the light would have to reflect in such a way that the angle of incidence is less than the angle of reflection. In Diagram B above, the eye is sighting along a line at a position below the actual image location. In this case, for light from the object to reflect off the mirror and travel to the eye, the light would have to reflect in such a way that the angle of incidence is more than the angle of reflection. Neither of these cases would follow the law of reflection. In fact, in each case, the image is not seen when sighting along the indicated line of sight. It is because of the law of reflection that an eye must sight at the image location in order to see the image of an object in a mirror.
Check Your Understanding 1. Consider the diagram at the right. Which one of the angles (A, B, C, or D) is the angle of incidence? ______ Which one of the angles is the angle of reflection? ______
Answer: Angle B is the angle of incidence (angle between the incident ray and the normal). Angle C is the angle of reflection (angle between the reflected ray and the normal).
2. A ray of light is incident towards a plane mirror at an angle of 30-degrees with the mirror surface. What will be the angle of reflection?
Answer: The angle of reflection is 60 degrees. (Note that the angle of incidence is not 30 degrees; it is 60 degrees since the angle of incidence is measured between the incident ray and the normal.)
3. Perhaps you have observed the image of the sun in the windows of distant buildings near the time that the sun is rising or setting. However, the image of the sun is not seen in the windows of distant building during midday. Use the diagram below to explain, drawing appropriate light rays on the diagram.
Answer: The ray of light drawn from the sun’s position at 7pm to the distant window reflects off the window and travels to the observer’s eye. On the other hand, a ray of light drawn from the 1pm sun position to the window will reflect and travel to the ground, never making it to the distant observer’s eye.
4. A ray of light is approaching a set of three mirrors as shown in the diagram. The light ray is approaching the first mirror at an angle of 45-degrees with the mirror surface. Trace the path of the light ray as it bounces off the mirror. Continue tracing the ray until it finally exits from the mirror system. How many times will the ray reflect before it finally exits?
Answer: The light reflects twice before it finally exits the system. Draw a normal at the point of incidence to the first mirror; measure the angle of incidence (45 degrees); then draw a reflected ray at 45 degrees from the normal. Repeat the process for the second mirror.
FUN FACTS: SPEED OF LIGHT in air was measured accurately by Albert Michelson, who won the Nobel prize in 1907 for his work. Using a rotating eight-sided mirror, he measured the time light took to make a round-trip between mountain tops that were a known distance apart.