1 Introduction
This report covers the current the Ontario curriculum’s sections regarding Fermat’s Law and its derivatives. The curriculum covers a simple portion of Fermat’s law fit for the understanding of high school students. In the report, both mentions of Fermat’s law and Fermat’s principle are referencing the same thing, the contribution Fermat has made to the world and it’s understanding of light and how it behaves. Having children gain interest in complex concepts such as the behaviour of light is important to bringing a new generation of scientific thinking individuals whom will contribute in making humanity a more advanced race. Photonics has high level applications in almost all fields. There is also high demand for experts regarding light in Ontario[1]. Concepts from Fermat’s work are covered in grade 9 through 12. The principles range from simple geometry of light paths and definitions to advanced multilayered lenses and polarization angles. The report will go over what is currently taught, when it is taught, how it is currently taught, and how it can be taught better to spark interest and fascination in adolescents and young adults.
2 Overview of {Instructional Topic}
Fermat’s principle explains the way light moves from one point to another, and its derivatives. This involves the refraction and reflection of light. The principle he is known for is the “least time” principle [2]. This principle uses trigonometry to get the distances between two points during a reflection of light. Using this, the time spent travelling can be found. The rule is that the light will always take the path which requires the smallest amount of time[2]. Fermat’s law’s derivatives can explain concepts of refraction and other optics mediums [3].
Currently, the Ontario curriculum only covers the reflection angles portion of Fermat’s law, putting aside most of the complex deriving to make it understandable to student in grades 9 and 10 [4]. Under the optics portion of grade 9 and 10 science, the definitions and geometry of light is studied and understood [4]. Students are exposed to the behaviours of light and receive a general understanding of how to complete calculations regarding angles of reflection. In grade 12 physics, Snell’s law and derivatives of Fermat’s law regarding refraction are taught. This contains questions with layers of different mediums being reflected, needing for the application of both Fermat’s principle and Snell’s law. However, grade 12 physics mostly focuses on the way light acts as a wave[5]. The bulk of Fermat’s law is still embedded in grades 9 and 10 in the geometry of light portion.
Many devices in the market use optics to varying degrees. These technologies include but are not limited to, screens, flashlights, x-ray scanners, and eye glasses. Improvement on these devices is important for a more advanced tomorrow as it give humans more ease as they attack more severe problems. Ontario has a very high demand for people who have an advanced understanding in the behaviours of light [1]. The way to deal with this high demand is to expose kids to optics and light at a young age, high school being ideal as this is where adolescents are first being exposed to more advance concepts [1]. With interest in the topic they can go on to fill the many roles these experts play in society. These jobs vary in technology to education planning. Space is also a field where humans are looking at going next, and optics experts are needed to study polarization angles and the effects of direct sunlight on space ships to give astronauts and satellites safe passage way to space and have them safely stay there [6].
3 Theory of {Instructional Topic}
Fermat’s principle refers to Fermat’s work in light and the way it behaves. He was a mathematician. Fermat could explain mirages and curved optic problems using his work, things that Snell’s law was not able to explain[2]. Dissatisfied with Descarte’s proof, Fermat tried another approach to refraction. Using Pythagorean’s theorem Fermat could derive a way to find the time and distances in such problems. Fermat uses the “least time” principle to explain how light will always try to find the fastest route to get from one point to another [3]. In high school, the portion covered is just reflection.
[7] (Reflection principle)
The angle of incidence (Øi in figure 1) and the angle of reflection are equal when a light is reflected on a surface [7]. Calculations can go both ways, through the incidence and the reflection angles. The normal line is perpendicular to the reflective surface. Using the trigonometry in this the distances between points can be found. Questions about mirrors and distances can be solved this way. The definitions are important for students to understand. The angle of incidence and reflection is the angle between the normal and the respective light paths. A mistake would be to assume the angle between the incident path of light and the reflective surface is the angle of incidence.
Applications of optics in University would be in the more complex derivatives of Fermat’s law where refraction and other paths of light are explained. This would be found in purely physics courses. The University of Western Ontario offers physics 3380 where Optics and Photonics are taught to a higher level degree[8]. This course contains all reflection and refraction content from high school and touches on new concepts for students such as polarization and Plasmonics [8]. This would have applications in technologies for telescopes and cameras, especially on the space front.
4 Existing Instructional Approaches
In textbooks, there are many examples of how the concept of reflective angles are taught. One good way is to look at the game Billiards. In Billiards players must hit different coloured balls to get them into holes. To do this, players must aim the ball carefully to get the correct angles. Much like light the incident and reflective angles are the same [9]. The angles and distances the ball travels can be measured which helps students gain a deeper understanding of the working of light. As light cannot be observed in this way as easily. The problem with this approach is that most schools do not have a pool table, though the purchase of one would have many applications in both entertainment and education.
The concept can also be demonstrated by using a plain mirror to reflect the photons from a laser. If done flat on the base, the light is viewed as a line on the base and the angles are made obvious and can even be measured. A normal line is drawn perpendicular to the mirror and the results are observed [9]. This is another great way of visualizing the behaviour of light and how it travels. The actual photons are moving much too fast to be observed effectively but the path is obvious. The only materials needed are a piece of white paper, a laser pointer, and a mirror. These are all cheap and easy to acquire.
The demonstration for waves can be done by the teacher with a laser and slits, performing slit experiments. Students can be given polarized lenses to view light, observing what they can and can’t see. As perpendicular waves of light will not be visible and are blocked by the lenses. This was done at A.B. Lucas high school in 2018 for the grade 12s. The instructor would take a laser and aim it through different sized slits and lenses, causing different patterns of dots on a wall. The distances between the lens and the laser, the lens and the wall, and distance between the dots would be taken for calculations. Also, a multilayered reflective surface will have a laser pointed at it. The way the light travels through or is reflected is observed, as the different layers may have different refraction indexes and the angle of incidence may not be equal to the angle of reflection. Both approaches are slightly more expensive to set up but a one-time purchase is all it takes for many years’ worth of demonstrations. A knowledgeable instructor can set it up for the class to see and demonstrate the more advanced portions of high school optics.
Illustrating ray diagrams can be done by students to increase understanding. Ray diagrams show how only some parts of things are visible in mirrors at one moment for one point of view. This is due to the nature of light reflection. Light will always find the shortest path from one point to another unless affected by the gravity of a black hole. To draw the diagram, a line is drawn in the middle of a page, and from one of the extreme points a point is drawn. Then, using bold line for the reflected light and dotted line for the light seemingly behind the mirror a path of light after reflection is drawn.
(Figure 2)
Then, the incident ray is drawn, without angles the line can be drawn from the position where the light contacts the mirror the opposite point of the light inside the mirror.
(Figure 3)
Finally, draw in the rest of the extremities which are in the last corners of the diagram away from the observer.
(Figure 4)
This approach gives a good understanding of how the light in mirrors are reflected and highlights the field of view one may have[10]. A cheap and effective way of teaching students about reflection, without worrying about angles or distances.
An alternative way for students to learn about light is with clear jelly. With molds of different depths, they can make different thickness jellies. With a laser pointer, they can point it at a consistent angle and observe the change in path when different thicknesses are used[11]. This is a cheap way for students to understand the way refraction works and the index of refraction can be calculated with the information retrieved. The distances the dot moves when the thickness is changed by a specific amount. This approach may cause a mess which is why schools tend to not go with this approach as food is involved. It would also be wise to not let young kids do this as they may eat contaminated jelly and get sick, causing concerns for school administrations.
The main goal is to get students to gain a general understanding for how the geometry of light works and how to calculate the distances between points. If this can be visualized than the process would have been a success. Another important factor is to have the students gain interest in the topic, which is why the more interactive or fascinating approaches such as the slit experiment are popular. Having students gain interest will cause more of them to enter the field of photonics, which is in high demand in Canada.
Table 1: Identified Instructional Aids
Instructional Aid
Source and Cost
Technology Basis/Bases
Strengths
Weaknesses
Billiards table
Quite expensive though some schools have one already
Very low
Good visualization of light reflection.
Balls are stunted by friction while light is not.
Laser and Mirror
Cheap
Not advanced
Good visualization of light reflection.
Polarized lenses and lasers
Expensive
High, nanotech required
Interesting to look at, good visualization of wave nature and refraction.
Jelly powder and water, laser or flash light
Cheap and bought from the store
Low
See refraction and reflection.
Hard to make clear jelly.
5 Applicable Technologies
There are a few technologies available for optics education in high school. For optics demonstrations, there are just a few things necessary. A source of light, and a way of manipulating it. This could be done by reflecting it, refracting it, or blocking it.
A technology that uses this would be telescopes. Specifically, Newtonian telescopes. Newtonian telescopes use mirrors to redirect light onto a small eye piece where a lens is then attached for the viewer to observe[12]. Students could try to measure magnification of different curved mirrors. The lens could also be manipulated and have different thicknesses. With both these variables on can be kept constant while the other is incrementally changes, with an observation at each stage to see how the magnification will change. This instructional approach would be expensive as the mirrors would have to be perfectly curved and made for clear viewing. A machine would also have to be made to switch out the curved mirror portion of the machine and allow for different sized curved mirrors to fit. That is if the lens on the eye piece is what is being constant. If the mirror is kept constant, changing the eye piece is a lot simpler as most telescopes already operate in this manner. The curvature of the mirror can be calculated if the lens size and the magnification is known.
(Figure 5)
The situation in figure 5 is reminiscent of the Ray diagrams from earlier. Using telescopes would be a great way of teaching about refraction and reflection[13]. The most ideal way to do something like this would be to make a small-scale telescope where the mirror can be switched. This would make it cheaper as there is no need for precise measurements and students could use the device freely without fear of breaking an expensive piece of equipment. Curved mirrors and lenses on their own is much cheaper than buying a telescope. This would be an interesting way of having students apply optics and reflecting angles to a real-world technology.
Another way students can be exposed to Fermat’s law is through mirror systems. Previous examples have used mirrors and lasers but another approach could be more variable and helpful for the student. Have a setup of 3 mirrors and have a final point which a laser needs to hit. Have the laser in a fixed position but the mirrors able to be turned and moved. The student then must turn them accordingly and work from the first mirror to the last one to get the right angles of incidence. Then the laser would be turned on and the more accurate the student is the better they applied their knowledge. This way the student not only observes the concept, the student must actively apply it. This device would be cheap to make as the components of mirrors and lasers are easy to get from any store. The device is simple to use and instruct and offers a quick look at reflection angles. It is suggested that this is used initially before the optics unit in grade 9 or 10 so the students can be tested for optics knowledge and so their fascination towards Fermat’s law will be sparked[4].
The best option for scale and third world countries would be the cheaper option which is the mirror systems device. Laser pointers and mirrors can be cheaply shipped out for these countries for them to quickly inspire a new generation of scientifically fascinated students. For the 1050 project, the mirrors option would be the most preferable due to the low costs and simplicity of use, without compromising the knowledge that is both obtainable and retainable.
Table 2: Potentially Applicable Technologies
Technology
Source and Cost
Potential Role in Instructional Aid
Strengths for Use in a Teaching Aid
Weaknesses for Use in a Teaching Aid
Telescope
Expensive
Observations of magnification
-Cool technology to spark interest
-Measurable advanced curvature of mirrors (calculated from lens size and magnification)
-Hard to use for beginners
-Hard to make
Mirrors
cheap
Angles of Incidence knowledge
-Good interactive experience,
-Learning about reflection angles
-Easy to use and instruct
-Fast demonstration of light reflection angles