Engaging students: Deriving the Pythagorean theorem

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Gary Sin. His topic, from Geometry: deriving the Pythagorean theorem.

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How could you as a teacher create an activity or project that involves your topic?

The Pythagorean Theorem is an extremely important topic in mathematics that is useful even when after the students graduate high school and proceed to college. As a student majoring in mathematics, I always like to explore the fundamental proofs of different theorems; I feel that if the student is able to derive a formula or theorem; it displays mastery over a mathematical topic.

As such, I will have the students work with a geometrical proof of the theorem. The students will be given 4 triangles with sides a, b, and c, and a square with sides c. I will instruct the students to fidget with the shapes and allow them to explore the different combinations that might lead to the theorem. As the class slowly figures out what combinations work, I will provide algebraic hints to the proof of the theorem. (including (a+b)^2 and c^2).

Finally, once a majority of the students figure out the geometric proof of the theorem; I will recap and reiterate the different findings of the students and summarize the geometric proof of the theorem.

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How can this topic be used in your students’ future courses in mathematics or science?

Pythagorean Theorem is extremely useful when beginning geometry, it applies to all right triangles and one could use it too to find the area of regular polgyons as they are also made up of right triangles. The surface area and volumes of pyramids, triangular prisms also rely on the theorem. Another major topic in geometry is trigonometry, where the trigonometric ratios are introduced and they are also based on right triangles. The Law of Cosines is also derived from the theorem. The theorem is also used in the distance formula between 2 points on the Cartesian plane.

The theorem is also used in Pre-Calculus and Calculus. Complex numbers uses it (similar to the distance formula). The basis of the unit circle and converting Cartesian coordinates to polar coordinates or vice versa also utilizes the theorem. The fundamental trigonometric identity is also derived from the theorem. Cross products of vectors uses the theorem, the theorem can also be seen in Calculus 3 in 3 dimensional geometry and finding volumes of various shapes because the theorem still applies to planes.

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How does this topic extend what your students’ should have learned in previous courses?

The theorem uses algebra to represent unknown sides in a right triangle. The students should have also learned about the names of the different sides on a right triangle, namely the legs and the hypotenuse. Being able to identify which side is the hypotenuse is very important in understanding and applying this theorem. Additionally, the students must be able to recognize what a right angle is which will determine if a triangle is a right triangle or not.

Deriving the theorem requires knowledge on the multiplication of polynomials, and how they are factored out. The students also use powers of 2 in the theorem and should be aware of how to square 2 integers and what the product is equal to. In the case of a non Pythagorean triple, the student must be able to manipulate radicals and simplify them accordingly.

Finally, the student must be able to identify what variables are provided and know what unknown they have to solve for. The variables and unknown side requires basic knowledge on how algebra works and how to use equations and manipulate them accordingly to solve for an unknown.

Engaging students: Using the undefined terms of points, line and plane

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Alizee Garcia. Her topic, from Geometry: using the undefined terms of points, line and plane.

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How could you as a teacher create an activity or project that involves your topic?

There is various way I could create an activity for this topic, but I think one that would be the most successful a project for the students in which they can better understand the terms. Since all three terms are related and relatively simple to describe the project could also be an in-class activity depending on the time given. However, in this project the students would have to take pictures of real-world examples for a point, line, and plane as best as they can and describe why they chose the examples they did. It is important that when teaching geometry as well as other lessons, that real-world examples are given to help students better understand the topics. Also, students can give their best definitions of the terms as well as drawing out them. This will allow students to think about the terms mathematically and as real-world subjects too.

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How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The use of undefined terms point, line and plane can be used in video games such as Minecraft and call of duty. Both games consist of a map of some sort with different coordinates of safe zones or just where the game will take place. In call of duty, using an aiming weapon allows for the player to find a point and from there to where they are aiming from is the line that will connect it. As well as in Minecraft, you are able to build off of other buildings as well as being able to connect the points in a certain grid in order to succeed. I think video games and technology would be the most common pop culture examples that this topic will appear in. Although there are far more video games that relate to the undefined terms of point, line, and plane, it is a good way to let students understand how geometry can be seen in the real world.

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How was this topic adopted by the mathematical community?

The undefined terms point, line and plane, are based off Euclidean geometry, which was brought up from Euclid of Alexandria, a Greek mathematician. This topic of the undefined terms point, line, and plane were discovered after the non-Euclidean was discovered. The topic of part of Euclidean geometry which is the mathematical system that proposing theories based off of other small axioms in which these are those small axioms. These terms are considered undefined due to the fact that they are used to create more complex definitions and although they can be described they do not have a formal definition.  Euclidean geometry was said to be the most obvious that theories brought from it were able to be assumed true. Although this is not what makes up the entire Euclidean geometry, it is what is able to allow these terms to be undefined and furthermore used to define more complex terms.

References:

Artmann, Benno. Euclidean Geometry. 10 Sept. 2020, http://www.britannica.com/science/Euclidean-geometry.

Engaging students: Introducing the parallel postulate

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Enrique Alegria. His topic, from Geometry: introducing the parallel postulate.

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The parallel postulate dates back to a man named Pythagoras of Samos. Pythagoras was a Greek philosopher that created a mysterious cult, the Pythagoreans. The purpose of the cult was to seek out a universal truth about numbers and shapes and became the foundation for Geometry. “The Pythagoreans concluded that the one universal quality of all things in the universe, the one thing that everything had in common, was that it was numerable and could be counted.” (Bryan 2014). Improving the work of Pythagoras and other mathematician predecessors was a man named Euclid who originated from ancient Greece. It was through Pythagoras’s key teachings, such as the Pythagorean Theorem, that began the fundamentals of Geometry.

Euclid wrote thirteen books named the Elements. These books were the entirety of Geometry. The Elements starts with a few simple definitions and postulates that were to be built off of each other to prove propositions. Through that work, Euclid changed the world. A masterpiece of logical thought and deductive reasoning.

Euclid caused controversy for years and years to come due to a specific part from the Elements. The parallel postulate which states, “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Because this postulate makes drastic assumptions it is almost impossible to be proven. For that reason, the parallel postulate has caused so much controversy over the years. Euclid tried to prove all that he could without the parallel postulate and reached Proposition 29 of Book I. This topic further developed as mathematicians believed that the statement could not hold true. From there, several mathematicians are to follow on proving the Parallel Postulate.

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How did people’s conception of this topic change over time?

Over time the conception of the parallel postulate changed as many mathematicians tried to prove the postulate. Mathematicians wanted to prove that the postulate was not so much a postulate but a theorem. Several proofs were created, but none had succeeded in proving the postulate from the plane in Euclidean Geometry. As no mathematicians were able to do so they moved towards other dimensions or geometries.

The beginning of Non-Euclidean Geometries. Using the first four postulates of Euclid but create a new definition for the parallel postulate. For example, Nikolay Ivanovich Lobachevsky and János Bolyai were two mathematicians that held all postulates true but the parallel postulate true when discovering Hyperbolic Geometry. The parallel postulate has been modified as such, “For any infinite straight line  and any point  not on it, there are many other infinitely extending straight lines that pass through  and which do not intersect .” (Weisstein) This also led French mathematician Henri Poincaré to show the Hyperbolic Geometry was consistent through the half-plane model.

Many more geometries were able to follow a similar format of creating a parallel postulate equivalent to Euclid’s parallel postulate. “The parallel postulate is equivalent to the equidistance postulatePlayfair’s axiomProclus’ axiom, the triangle postulate, and the Pythagorean theorem.” (Szudzik). Despite the many trial and errors of trying to prove the parallel postulate, peoples’ conception of the topic was able to transform and discover new geometries where the respective parallel postulate can hold to be true.

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How can technology be used to effectively engage students with this topic?

Technology can be used to effectively engage students with the parallel postulate through a short series of YouTube videos by the channel Extra Credits. The five-part video series is called “Extra History: History of Non-Euclidean Geometry” with short seven to eight-minute videos which goes through the history of the parallel postulate. The video not only explicitly states what the parallel postulate is, but it goes through the history of how peoples’ conception has changed over time and how it has applied to today’s world and expands into physics.

The video series is produced with high-quality animation and narration. An engaging visual representation of the history of geometry that mathematicians have gone through to prove Euclid’s parallel postulate. Engaging in the countless trials and the amount of time that it has taken to go through this proof. Showcasing other discoveries that Euclidean Geometry has led to being Non-Euclidean Geometry. Lastly, the discoveries that Non-Euclidean Geometries will further lead to. Allowing students to join in on the questioning of the world as we know it.

Citations

Bryan, V., 2014. The Cult Of Pythagoras. [online] Classical Wisdom Weekly. https://classicalwisdom.com/philosophy/cult-of-pythagoras/

Szudzik, Matthew and Weisstein, Eric W. “Parallel Postulate.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ParallelPostulate.html

Weisstein, Eric W. “Non-Euclidean Geometry.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/Non-EuclideanGeometry.html

https://mathcs.clarku.edu/~djoyce/java/elements/bookI/post5.html

https://www.youtube.com/watch?v=nkvVR-sKJT8&list=PLhyKYa0YJ_5Dj3ZG-Qk9VfaCfo-Nh9S-2

Engaging students: Finding the area of a square or rectangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Austin Stone. His topic, from Geometry: finding the area of a square or rectangle.

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How could you as a teacher create an activity or project that involves your topic?

There are many applications to the real world that involves geometry and specifically area of squares and rectangles. Students could use this topic to find the cheapest cost of tiling the floor of a bathroom. Giving them the dimensions of the different tiles and the cost of each tile, students would have to find the area of the bathroom floor and then be able to pick the set of tiles that would be the most efficient and cheapest. This gives students a real world application to what they are learning while also giving them practice in finding the area given dimensions of a square and/or rectangle. This project also calls back to prior knowledge such as perimeter of rectangles and multiplying cost of one tile with the number of tiles used to get to total price. This project could also be a small part of a bigger PBL using area and perimeter of multiple polygons.

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How does this topic extend what your students should have learned in previous courses?

The obvious prior knowledge to finding the area of a square of rectangle is being able multiply two numbers which is learned back in grade school. If the students are given the area of the square or rectangle and labeling the sides with a variable, the students would have to be able to solve for the variable. By doing this they would have to be able to multiply binomials (or polynomials if you want students to have more of a challenge). Once they multiply the two binomials and set the equation equal to the area given, they would then have to use the quadratic formula or factor which is learned in Algebra I. If students are given one side and the area, then they would have to solve for a variable with degree one which is used continually in all math classes. Depending on what information is given in the area problem, students will have to use prior knowledge to determine the answer.

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How have different cultures throughout time used this topic in their society?

In East Asian mathematics during the 1st-7th centuries, a book called The Nine Chapters gives formulas for solid figures including squares and rectangles. The formulas are given as series of operations to get the result, called algorithms. Instead of variable and symbols, the formulas are given in sentences as in, “multiply the length of the rectangle by the width.” This puts the regular A=lw into words so that if someone who had no idea how to compute the area, they would be able to understand by the sentence given. This undoubtably was much more difficult to follow and became too long of descriptions for more complex figures, as this way of mathematics ended in Eastern Asian in the 7th century. That does not mean that this way of math was not important. This put words into formulas instead of symbols which made it easier to understand for those that are learning it for the first time.

References

https://www.britannica.com/science/East-Asian-mathematics/The-great-early-period-1st-7th-centuries

Engaging students: Finding the volume and surface area of prisms and cylinders

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Angelica Albarracin. Her topic, from Geometry: finding the volume and surface area of prisms and cylinders.

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How could you as a teacher create an activity or project that involves your topic?

For finding the surface area of prisms and cylinders, I as the teacher would create an activity centered around using the nets of these figures to better visualize this concept. In my experience, many students do not struggle with the computational aspect of finding the surface area of prisms and cylinders, but rather, they tend to forget to calculate the area of all the faces of such figures. When a student views these three-dimensional figures on paper, it can be easy to forget some faces as not all of them can be illustrated, requiring the student to have an accurate depiction of the figure already in mind. By having students work with nets, they will have some guidance in calculating the surface area of prisms and cylinders. Additionally, having the students construct each intended figure with the net can also help students develop a better understanding of the composition of prisms and cylinders.

A project I could use as a teacher in order to help students understand volume of prisms and cylinders would be to have the students create their own drink company. I could provide the students with several models of different styles of cans they could use and have them find the volume of their selected can as a requirement. I think this would be a fun way to not only allow to students some creative freedom but also provide practice calculating the volumes of various prisms and cylinders. Students would have to consider aspects such as how much liquid one container holds over another, how portable the shape is, and how will others drink from it. Students could also find the surface area of their drink cans in order to see how much material would be needed to print a label that would fit around each can.

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How can this topic be used in your students’ future courses in mathematics or science?

Finding the volume and surface area of prisms and cylinders provides a basic background for students to start exploring more complex shapes such as spheres, cones, and pyramids. However, in Calculus I, this topic is taken further with the introduction of integrals and the concept of finding the area under irregular curves. Later down the line, students will also learn about volumes of solids of revolution. For rounded curves, an approximation for such solids is comprised of taking the sum of the volume of many cylinders; the more cylinders there are, the closer the approximation will be to the true volume. An image of this is shown below:

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Continuing with the theme of solids of revolutions, Calculus II is when students must find the surface area of these solids. To approximate the surface area, we take the surface area of frustums that can be formed under the curve. Frustums are similar to cones as they both have circular bases, but instead of coming to a point, a frustum also has a circular top. As before, the greater the amount of frustums used in the approximation, the closer the calculated value is to the true surface area. The formula for the surface area of a frustum is A = 2\pi r h A = where r =(r_1+r_2)/2. Frustums are unique in that both circular bases are different. In the case that the bases are the same, the formula for r becomes r =(2r_1)/2 = r_1,  in which case the formula for surface area becomes A = 2\pi r h  which is exactly the formula for the surface area of a cylinder. Below is an image of the surface area approximation of a solid formed by revolution:

This image has an empty alt attribute; its file name is cylinder2.png

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

The ancient Greeks are responsible for naming many of the figures and solids we commonly see in Geometry. For example, the word “prism” comes from the Greek word meaning “to saw”, which comes from the fact the cross sections (or cuts) of a prism are congruent. The word “cylinder” also comes from Greek, specifically from the word that means “to roll”. In addition, the Greeks were also “the first to systematically investigate the areas and volumes of plan figures and solids”. One of the most famous of these Greeks is the mathematician Archimedes who is directly responsible for the approximation of the area of a circle, the approximation of pi, the formulas for the volume and surface area of a sphere, and a technique called the “method of exhaustion”, which was used to find areas and volumes of figures in a manner similar to that of modern calculus. Archimedes viewed his discovery of the formula for the surface area of a sphere as his greatest mathematical achievement and even instructed that it be remembered on his gravestone as a sphere within a cylinder.

Another mathematician who developed techniques that bore similarities to modern calculus was Italian mathematician Bonaventura Francesco Cavalieri. While his discoveries pertained to finding the volume of objects, he was able to use are of cross sections to show that “two objects have the same volume if the areas of their corresponding cross-sections are equal in all cases”. This came to be known as Cavalieri’s Principle, but it is important to note that Chinese mathematician Zu Gengzhi had previously discovered this principle hundreds of years before Cavalieri. The next biggest advancement in this topic is attributed to integrals and making sense of the idea of finding the area under a curve. An approximate method for finding the area of a figure with an irregular boundary was developed known as Simpson’s Rule which had previously been known by Cavalieri but was rediscovered in the 1600s.

References:

https://amsi.org.au/teacher_modules/area_volume_surface_area.html

https://www.famousscientists.org/archimedes-makes-his-greatest-discovery/#:~:text=Archimedes%20also%20proved%20that%20the,a%20sphere%20within%20a%20cylinder.&text=The%20sphere%20within%20the%20cylinder.

https://study.com/academy/lesson/how-to-find-the-volume-of-a-cylinder-lesson-for-kids.html

https://tutorial.math.lamar.edu/classes/calci/Area_Volume_Formulas.aspx

https://tutorial.math.lamar.edu/classes/calcii/surfacearea.aspx

https://en.wikipedia.org/wiki/Surface_area

A line joining two infinitely small points

Been there, done that.

Source: Math With Bad Drawings, https://www.facebook.com/MathWithBadDrawings/photos/a.822582787758549/3862434577106673/

Happy Pythagoras Day!

Happy Pythagoras Day! Today is 12/16/20 (or 16/12/20 in other parts of the world), and 12^2 + 16^2 = 20^2.

righttriangle121620

Engaging students: Using right-triangle trigonometry

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Cody Luttrell. His topic, from Precalculus: using right-triangle trigonometry.

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A.1 Now that students are able to use right triangle trigonometry, there is many things that they can do. For example, they know how to take the height of buildings if needed. If they are standing 45 feet away from a building and they have to look up approximately 60 degrees to see the top of the building, they can approximate the height of the building by using what they know about right triangle trigonometry. Ideally, they would say that the tan(60 degree)= (Height of building)/(distance from building = 45). They can now solve for the height of the building. The students could also use right triangle trigonometry to solve for the elevation it takes to look at the top of a building if they know the distance they are from the building and the height of the building. It would be set up as the previous example, but the students would be using inverse cosine to solve for the elevation.

 

 

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A.2 An engaging activity and/or project I could do would be to find the height of a pump launch rocket. Let’s say I can find a rocket that states that it can travel up to 50 feet into the air. I could pose this problem to my students and ask how we can test to see if that is true. Some students may guess and say by using a measuring tape, ladder, etc. to measure the height of the rocket. I would then introduce right triangle trigonometry to the students. After a couple of days of practice, we can come back to the question of the height of the rocket. I could ask how the students could find the height of the rocket by using what we have just learned. Ideally, I would want to here that we can use tangent to find the height of the rocket. By using altimeters, I would then have the students stand at different distances from the rocket and measure the altitude. They would then compute the height of the rocket.

 

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D.1 In the late 6th century BC, the Greek mathematician Pythagoras gave us the Pythagorean Theorem. This states that in a right triangle, the distance of the two legs of a right triangle squared added together is equal to the distance of the hypotenuse squared (a^2+b^2=c^2). This actually was a special case for the law of cosines (c^2=a^2+b^2-2ab\cos(\theta)). By also just knowing 2 side lengths of a right triangle, one may use the Pythagorean Theorem to solve for the third side which will then in return be able to give you the six trigonometric values for a right triangle. The Pythagorean Theorem also contributes to one of the most know trigonometric identities, \sin^2 x+\cos^2 x=1. This can be seen in the unit circle where the legs of the right triangle are \sin x and \cos x and the hypotenuse is 1 unit long. Because Pythagoras gave us the Pythagorean Theorem, we were then able to solve more complex problems by using right triangle trigonometry.

 

Engaging students: Introducing the parallel postulate

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Eduardo Torres Manzanarez. His topic, from Geometry: introducing the parallel postulate.

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A2) How could you as a teacher create an activity or project that involves your topic?

The Parallel Postulate is an interesting statement that intertwines line segments and angles. This postulate states that if a straight line intersects two straight lines and the interior angles on the same side add to less than 180 degrees, then those two straight lines will intersect on that side if the lines are extended. Simply, if a straight line intersects two other straight lines and the interior angles on the same side add up to 180 degrees then the two lines are parallel. One activity that can get students to understand this axiom how test the validity would be to provide sets of straight-line segments and ask students to form interior angles and find their measurements. This would be particularly best to be done with technology such as a software like GeoGebra. Students would be given a set of line segments. First, provide nonparallel line segments such as the ones below.Next, ask students to draw any line segment such that it intersects the two previously given. Letting students make their own particular line segment can suggest that the validity of the statement is universally true.

Now students can use the angle tool to measure the interior angles on both sides. The pictures below are an example.

So, in this example, the right-side interior angles add up to less than 180 degrees and so the given two lines will intersect on the right side. Students can check that the lines segments intersect by placing lines over these segments and check for an intersection. The following image provides evidence as to this being the case for the example.

Hence, this example shows some truth to the postulate. This activity can be further enhanced and propelled by giving students lines that are already parallel and checking any set of interior angles made by a third line segment. Students will find that any segment created will result in the interior angles on both sides to add up to 180 degrees exactly. Such an activity like this would be useful as an introduction to the Parallel Postulate.

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D1) What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid, a Greek mathematician, came up with the Parallel Postulate in his discourse titled Elements which was published in 300 BC. Elements is made up of 13 books that contain definitions, theorems, postulates, and proofs that make up Euclidean Geometry. The reason Euclid wanted to accomplish this was to ascertain all of geometry under the same set of axioms or rules so that everything was related to one another. Euclid’s accomplishment in doing this has resulted in him being referenced as the “Father of Geometry”. There is not that much information on Euclid’s life from historical contexts, but he did leave an extensive amount of work that propagated many fields in math such as conics, spherical geometry, and number theory. Elements is estimated to have the greatest number of editions, second to the Bible. The Parallel Postulate by Euclid led to many mathematicians in the 19th century to develop equivalent statements within the contexts of other geometries. Hence Euclid was able to propagate geometry even further, way after he passed away.

 

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D2) How was this topic adopted by the mathematical community?

Ever since Elements was made known through the mathematical community, many individuals tried to prove the Parallel Postulate by using the other four postulates Euclid wrote. There is evidence to suggest that Euclid only wrote this particular postulate when he could not continue with the rest of his writings. So, the mathematical community sought out to find a proof for it since the postulate was not clear to be trivially true, unlike the other postulates. Some mathematicians such as Playfair wanted to replace the Parallel Postulate with his own axiom. It was finally shown in 1868 that this postulate is independent of the others and therefore cannot be proven by the other postulates by Eugenio Beltrami. There has been development in a specific type of geometry known as absolute geometry which actually derives geometry without the Parallel Postulate or any other axiom that is equivalent to it. This shows how much the community has been up to challenging the postulate but also how to proceed without it to see if Euclid could have done the same.

 

Engaging students: Dilations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Andrew Sansom. His topic, from Geometry: identifying dilations.

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C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

In recent years, Marvel Studios’ Cinematic Universe films have exposed society to dilation. One of the beloved Avengers is Ant-Man, who starred in two of his own eponymous films, as well as in Captain America: Civil War and Avengers: Endgame. Ant-man is the hero identity of one Scott Lang, an engineer trying to be a good father for his daughter. In the process, he ends up associating with Hank Pym, who had developed a technology that make it possible to shrink and enlarge objects and people. In the aforementioned films, he utilizes this ability to solve problems and combat villains.

Two particular instances where he used this ability to shrink and enlarge in meaningful ways occur in Avengers: Endgame. One such moment is when Ant-man shrinks to a smaller size than that of an insect, and crawls inside of Tony Stark’s arc-reactor. He pulls apart one wire, which causes a short, and provides a long enough distraction for his team’s escape. Later in the film, after he and a few other Avengers were buried under a collapsed building, he dilates to a gargantuan size to push aside the rubble and rescue them.

 

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A2. How could you as a teacher create an activity or project that involves your topic?

Teachers could use this connection to Ant-man to their advantage by designing an activity where students must use geometric dilations to solve puzzles. Give the students several consecutive scenarios with diagrams and ask them to come up with a plan that Ant-man should follow to maneuver the course. In this plan, they must require at what locations Ant-man should dilate, and by what scale factor, then to where he should move to dilate again. To make this more puzzling, put another restriction on the course that it costs a certain amount of “Pym Particles” to run a distance while enlarged/shrunken or to do the shrinking in the first place. This encourages the students to minimize the dilations to reduce the cost.

Below is an (extremely rough draft) example level. Ant-man’s location is the square where his feet are. He must move right three squares. He must then dilate with a scale factor of 2, with his bottom right corner being the center of dilation. He then shrinks with a scale factor of ½ about his top left corner. He then moves right one square. He then shrinks with a scale factor of ½ about his top right corner. Then walk right 4 squares. He then expands with a scale factor of 2, shrinks with a scale factor of ½, walks right 2, expands, falls down one, then runs right.

This platformer puzzle could even be expanded into a video game of sorts maximum engagability.

 

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B1. How can this topic be used in your students’ future courses in mathematics or science?

Dilation appears in many topics in math later than geometry. Dilation is one of the major transformations studied in Algebra 2. Studies of geometric dilation will prepare students for analyzing how scale factors will stretch or compress functions. Furthermore, comfort in geometric transformations will prepare them for advanced integration problems. If students can identify the geometry of integral, then performing transformations, including dilation, can make certain problems easier to solve. In even further math classes, including linear algebra, scaling becomes an important tool in manipulating vectors. Students should realize at that point, that dilation is a certain type of linear transformation on a set of vectors representing a shape. The concept is also critical to an intuitive understanding of what eigenvectors are.