# A line joining two infinitely small points

Been there, done that.

# Happy Pythagoras Day!

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

# 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

# Engaging students: Midpoint

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 Cody Luttrell. His topic, from Geometry: deriving the term midpoint.

A1: Being able to find midpoint is a very important skill for students to learn in their geometry class. Some interesting word problems that students may be able to solve would being able to find the “half way point” between two locations. If they were wanting to meet their friend in a town that is equidistant from their town and their friend’s town, they may use midpoint to solve this. Other word problems may include running track, NASCAR, and can even be used in fast food examples. For example, Subway sells foot long sandwiches that are cut in half. How does the Subway worker know where to cut the sandwich where they have equal half’s? The student can find the midpoint of the distance of the sandwich and that is where they should make the cut. Knowing how to find midpoint will aid the student in the rest of their geometry class as well which can lead to more interesting word problems.

B1: Knowing how to find the midpoint between two points can greatly aid students in future subjects. One of the most common examples would be finding the vertex of a parabola. If the students looks at the x value for the roots of a quadratic, the student can find the midpoint between the two points which in results will give you the x value for the vertex since the function is even. This can then be applied to physics when dealing with projectiles. Students can find where an object reaches its maximum height if they know its starting point and landing point. The students then will also come across this topic when they get into calculus when they deal with integrals. Using Riemann sums end up using the midpoint formula to help estimate the area under the curve. As seen from above, midpoint can applied to many advanced mathematical or science courses that a student may be enrolled in.

C2: Knowing how to find the midpoint between two distances is used in art pieces and architecture around the world. To keep things symmetrical, one must know how to find the line of symmetry which is also the midpoint between the two points. Symmetry is used to make things appealing to the eye, which is a major concept of art in general. In architecture, having to know where the “middle” is located, is very important to keep things structurally sound. The reason for this, is if a building is weighted unevenly on opposite sides of the midpoint, it can create an unbalance which can end up being an unsafe environment. Knowing how to use midpoint can also be applied in theatre. The stage is divided up to left, center, and right stage. Finding the midpoint of the stage can help differentiate where center stage is compared to right and left stage.

# 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 Haley Higginbotham. Her topic, from Geometry: deriving the Pythagorean theorem.

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

An interesting hands-on activity would be to do a visual proof of Pythagorean Theorem by using just paper, scissors, a ruler, and a pencil. Starting with a square piece of paper, the students will make a square with a length in the bottom left corner and a square of b length in the upper right hand corner, similar to the picture below on the right-hand side. Then the students would cut out the squares, and end up with two squares and two rectangles. The students would then be instructed to cut the both rectangles along their diagonals. Then the challenge is to make a square that contains a square inside by only using the triangles they have cut out. The level of difficult of the challenge will depend on the grade level and on the caliber of students, but it’s still more interesting than writing out a formal proof. Then after everyone has made something similar to the picture below on the left-hand side, I would ask them if they know why this proves the Pythagorean Theorem. If a student has a good explanation, I would ask them to demonstrate their explanation to the rest of the class. If no one figures it out, I would suggest they label the different lengths and see if they figure it out then.

How has this topic appeared in high culture?

The Pythagoras tree is a fractal constructed using squares that are arranged to form right triangles. Fractals are very popular for use in art since the repetitive pattern is very aesthetically pleasing and fairly easy to replicate, especially using technology. The following picture is an example of a Pythagoras tree sculpture extended into 3 dimensions. There is also the Pythagorean snail, which is constructed by making isosceles right triangles in a circular pattern, keeping the smallest leg of each triangle the same size. With this basic design, you can create a variety of designs, an example is pictured below. Even though the base is a bunch of triangles in a spiral, the design overlaid on top of it takes it from purely mathematical to a piece of art. Of course, one could argue that mathematics itself is an art, but the general population would agree that the design really makes it a work of art.

How can technology be used?

I think I would use Desmos to extend the activity described in A2, since they have seen it works for their particular choice of a and b, but they might not see how it works for all choices of a and b (as long as the triangles they have are right triangles). By using Desmos, I can use an activity that allows students to drag the different sides to see that the relationship holds no matter how a and b change. I think something similar to this activity would work: https://teacher.desmos.com/activitybuilder/custom/5adc7bfced2ada678516940e, except I would modify it so it was closer to the activity that we did with paper during class. I would also show them the other explanation of the squares aligned with the lines of the triangles. It’s great because Desmos has activities that you can use but you can also customize the activities however you want to fit your specific ideas. You could also ‘code’ from scratch your own activity.

# Engaging students: Perimeters of polygons

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 Tiger Hersh. His topic, from Geometry: perimeters of polygons.

How has this topic appeared in pop culture?

In simulation games like ‘Farming Simulator’ and ‘Cities Skylines’ there is a need to determine the perimeter of the polygon or shape. When you first start out in the game you must keep in mind the limitations of your sandbox (which is usually in the confines of a square). This is where finding the shape of the polygon becomes very useful. When you are able to determine the boundary area of that square, you are then able to map out the land that you would wish to occupy. For Farming Simulator you can find how much space your plot of crops will take and how much space in between your fields.

In Cities Skylines (the city builder simulator) it is similar to that of Farming Simulator but instead of making separate plots of land for your crops, you are instead creating separate plots of land for living spaces, markets/businesses, and industrial spaces. The reason finding the perimeter of each type is important is so that you can create a space for them that is reasonable and does not intersect with the other districts. So it is key to having these each place separated but also given plenty of space, which would require you to find the length and width that would be appropriate to the land you have to work with and also for that place.

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

Once students have learned how to find the perimeter of a polygon they can use this concept and apply it to finding the distance between two points on a graph. Students may find the circumference of a circle (the perimeter of the circle) and use it on application problems that may come up in science; such as determine how far a person travels on a merry-go-round. This is then later used to determine circular motion or how fast something is going.

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

There are many interesting people/things that contributed to this concept. One notable person is Archimedes, the greek mathematician, who around 250 B.C. used the sides of hexagons to find an approximation of pi. The method was that he would have hexagons inscribed in the circle and then double the sides of the hexagon to a 12- sided polygon, doubled the sides again to a 24 sided polygon, doubled to a 48 sided polygon, and doubled again for a 96 sided polygon. He was then able to bring the perimeter of the polygons closer and closer to the circumference of the circle. This later turned out to be what we know now today as pi.

# Engaging students: Defining the term perpendicular bisector

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 Biviana Esparza. Her topic, from Geometry: defining the term perpendicular bisector.

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

A fun project that involves perpendicular bisectors is a project that I did in my project-based instruction class earlier this semester. The geometry project required students to create a piece of origami that had an angle bisector and a perpendicular bisector labeled. Leading up to the project creation and presentation day was a series of workshops and DIY activities in which students learned what terms such as congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, perpendicular bisector, circumcenter, and incenter. These activities included working with patty paper to create angle bisectors and perpendicular bisectors on triangles, worksheets where students had to graph triangles and find the circumcenter and incenter, independent practice, formative assessments, and lastly the final origami creation. It was fun to see students take ownership of their learning and be proud of their final origami creation, because they were allowed to create whatever they wanted as long as an angle bisector and a perpendicular bisector were labeled. Students had a firm understanding of what the key vocabulary terms were, especially perpendicular bisectors and angle bisectors, because they had used them so much throughout the workshops.

C1. How has this topic appeared in pop culture?

Paper Planes is an Australian film released in 2015 about a boy named Dylan who has a talent for making paper planes and wants to go the World Paper Plane Championship. In the movie, Dylan is taught how to make the “perfect” paper plane by a student teacher. Students start off by making simple planes like those that most people make. Although most people making paper airplanes don’t think of terms such as perpendicular bisectors or angle bisectors, they are the basics to making any form of paper airplane. The first step to making a plane, which is folding a piece of paper in half by aligning two opposite edges, creates a perpendicular bisector: the fold is a perpendicular bisector to the edges it touches. Students in a class learning about perpendicular bisectors could be shown minutes 5:40 to 8:21 to engage them about paper airplanes and they could be asked how paper planes could be related to geometry. This could show them that something as simple as a paper plane has some mathematical connections.

E1. How can technology be used to effectively engage students with this topic?

Gizmos is a website full of interactive simulations and lesson plans that effectively incorporate technology in the classroom. The website has a lesson titled “Segment and Angle Bisectors” in which students manipulate points to explore the properties of perpendicular bisectors and points on an angle bisector. This is a helpful tool to let students discover properties on their own instead of the teacher directly telling them what a perpendicular bisector is. The website also includes a worksheet with questions to go along with the gizmo exploration.

References

https://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=174

# Engaging students: Defining intersection

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 Marlene Diaz. Her topic, from Geometry: defining intersection.

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

Intersection is a term that the students will see for a very long time in math. There are many beneficial activities or projects that students can do that involves intersections. There project will include them to first define the term next find any examples where the term has been used. They will have to show two mathematical examples and five real world examples of intersection. For example, a student can compare a rail road intersecting and two lines intersecting on a graph. Furthermore, they will have to explain each image by answering questions like, how do you know it is intersecting, and can it intersect again. These questions can be answered for different examples. In conclusion, it will allow the student to connect a real-world example to a mathematical term. Since this is a very fast and small concept because we are just defining the term intersection, therefore I will consider this an activity the students can do in their group.

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.)

Girard Desargues was one of the people who contributed to the development of intersection. Desargues was born February 21, 1591 in France. He had significant contribution to mathematics, especially projective geometry. For part of his life he was very known, he met people like Rene Descartes and Pierre de Fermat who have also contributed to the mathematical community. In the 19th century his work was being rediscovered and resulted in the Desargues’s theorem. Desargues’s Theorem states that if two triangles ABC and A′B′C′, situated in three-dimensional space, are related to each other in such a way that they can be seen perceptively from one, then the points of intersection of corresponding sides all lie on one line, provided that no two corresponding sides are parallel. Furthermore, his best and most important work was from 1639 called, “Rough Draft of Attaining the Outcome of Intersecting a Cone with a Plane”. Although, he and his work were forgotten for a long time, he did help the mathematics community with all items rediscovered.

How has this topic appeared in the news?

One main thing that always appear on the news are traffic and traffic accidents. Some traffic is created when cars are attempting to cross an intersection. At most intersections there are traffic lights that only allow certain traffic to get across the intersection. Traffic lights are very helpful in intersections because it helps traffic get across without putting drivers in danger. The only time traffic accidents can happen at intersections is when a driver ignores the traffic signs, or they are at a stop sign. At stop signs in intersections not all four intersections will have stop signs therefore drivers should be careful in these areas. Intersections, like seen here, appear in the real world and they are something people don’t realize since its just a part of life, but if there are car accidents most happen near or at an intersection. These are some of the headlines some news report when car accidents, “Serious accident renews discussion about dangerous intersection” and “Motorcyclist killed in 5-vehicle crash at intersection with disabled traffic lights in Aurora”.

Citations

https://www.britannica.com/biography/Girard-Desargues