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.

 

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.

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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.

green lineB1: 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.

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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.

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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.

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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.

 

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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.

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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.

 

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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.

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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?

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.

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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.

 

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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.

 

 

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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://drive.google.com/file/d/1f3RuOWH6hIMROQIJgejJwuD6ngQWdZZ3/view?usp=sharing

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.

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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.

 

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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.)

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.

 

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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

https://www.athensnews.com/news/local/serious-accident-renews-discussion-about-dangerous-intersection/article_023e78e0-eb8a-11e9-bc93-2b11b4ffec8c.html

https://www.thedenverchannel.com/news/local-news/motorcyclist-killed-in-5-vehicle-crash-at-intersection-with-disabled-traffic-lights-in-aurora

 

 

Large number formats

A great explanation of the comic can be found at https://www.explainxkcd.com/wiki/index.php/2319:_Large_Number_Formats.

Engaging students: Graphing Sine and Cosine Functions

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 DeForest Mitchell. His topic, from Precalculus: graphing sine and cosine functions.

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C2: How has this topic appeared in high culture (art, classical music, theatre, etc.)?

There are multiple forms of art and music and theater using sine and cosine. I am going to focus on music. Tone and sound in itself is a reverb of sine functions with different wavelengths and amplitude. This is such a great importance not only in knowing sine functions but also being able to create music. Knowing what a tone is and why the sound is becoming higher/lower or quieter/louder. If the amplitude is shorter then the sound itself will be quieter and vise versa with the amplitude growing so will the over all sound. If the period is increasing or becoming longer, the sound will be deeper, while shorter periods will create sharper and higher pitch sounds. Once you combine the amplitude and the period in specific ways that’s how specific notes and tones are made. As shown below, there is a combination of amplitudes and periods to create a sound wave using sine functions. With this in mind mathematically you can predict what a tone or song will sound like just depending on the inflection of amplitude and frequency of the function.

 

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

One big component to my personal understanding of graphing sine and cosine is seeing a physical model of it. Such as a spring in physics, or the light spectrum. There are websites such as https://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html to be able to represent a sine function being sketched as a weighted spring bounces up and down. There are different slides to adjust the stiffness and dampening of the spring to show the alternate forms of the same graph. Since and cosine are used in many different forms of sciences, such as wavelengths (as shown in the picture below). This is a great way to show the students that there is a reason to learn the subject and that there are practical uses for it in life outside of school. For example, with the wavelength spectrum a teacher can make the correlation that a sine function with a longer period would be like soundwaves that have a longer wavelength, also to show that there are only certain wavelengths that we can see and there can be a correlation to graph the sine period and that if you were to make it into a function that there is only a certain few period lengths for a sine function for humans to be able to see colors.

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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? (You might want to consult Math Through The Ages.)

Circles have been around for millions of years and are to this day one of the simplest shapes. With Circles and by tracking the circumference of the circle you can make sine and cosine graphs. Joseph Fourier (1768 – 1830 A.D.) was a very influential person with the devilment of graphing sine and cosine functions. He was so due to his the foundning of the Fourier series. “Fourier series (thus the Fourier transform and finally the Discrete Fourier Transform) is our ancient desire to express everything in terms of circles or the most exceptionally simple and elegant abstract human construct. Most people prefer to say the same thing in a more ahistorical manner: to break a function into sines and cosines.” The summative way to say saying that Fourier came up with an equation to take any repeating series and be able to turn it into forms of sine and cosine to be able to graph. Which in turn creates a circle to better understand the said repeating series. This was a great mathematical find to show the correlation between any repeating series and the more well-known sine and cosine terms today. As shown below, with the Fourier series they were able to take a repeating series to “convert” them into a circle and then be able to graph the functions in terms of sine and cosine.

Resources:

https://www.gnu.org/software/gnuastro/manual/html_node/Fourier-series-historical-background.html

https://www.google.com/url?sa=i&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwjCt_3VjJXlAhVEQKwKHdm4CmcQjhx6BAgBEAI&url=https%3A%2F%2Fwww.extremetech.com%2Fextreme%2F252295-layered-solar-cell-can-capture-wavelengths-solar-spectrum&psig=AOvVaw1RQwNN_tnNutYxnp_BFrF4&ust=1570913952314160

https://www.google.com/url?sa=i&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwic7YDjk5XlAhURCawKHU_CCOEQjhx6BAgBEAI&url=https%3A%2F%2Filovefood1234.weebly.com%2Ffrequency-and-amplitude.html&psig=AOvVaw1f48xa071u-Hn-t1TIH4DN&ust=1570915856599588

https://www.google.com/url?sa=i&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwiL_-yxlZXlAhUKT6wKHUDbBJMQjhx6BAgBEAI&url=https%3A%2F%2Fwww.colourbox.com%2Fvector%2Fmusic-sound-waves-vector-27786168&psig=AOvVaw1YIMBZ1aL5pO2uOkl-ksg2&ust=1570916292788406

Engaging students: Finding the area of a right triangle

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 Cory. His topic, from Geometry: finding the area of a right triangle.

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

Finding the area of a right triangle opens up the door to all sorts of applications in the future. The next step is the Pythagorean theorem which is used constantly throughout many math courses. The study of right triangles also opens up the world of trigonometry with students will be using in nearly every math course they go on to take. Once knowledge is learned of right triangles, other triangles can be manipulated to look like right triangles, or to create right triangles within normal triangles. Triangles are even utilized when determining things about other shapes as well, such as dividing rectangles into 2 triangles and other manipulations. If they go on to pursue geometry further, the Pythagorean theorem is one of the first couple of theorems proved and used in book 1 of Euclid.

 

 

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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?

Pythagoras was a Greek philosopher that contributed to right triangles. He is credited with discovering possibly one of the most important right triangle properties. A legend says that after he discovered the Pythagorean theorem, he sacrificed an ox, or possibly an entire hecatomb, or 100 cattle, to the gods. The legitimacy of this legend is questioned because there is a widely held belief that he was against blood sacrifices. The Pythagorean theorem was known and used by Babylonians and Indians centuries before Pythagoras, but it is believed he was the first to introduce it to the Greeks. Some suggest that he was also the first to introduce a mathematical proof, however, some say this is implausible since he was never credited with proving any theorem in antiquity.

 

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

Applications such as Geogebra can be used for any type of geometry activity. It is a great way to help kids visualize what is happening with shapes in geometry, something that is usually a struggle for students. For helping students understand how to find the area of a right triangle, it can easily be shown that if you take a rectangle, or a square, and cut it in half diagonally, you get two right triangles. And since the area of a right triangle is half of the area of a rectangle or square. The various ways that shapes can be manipulated virtually can be a big help for students that learn in different ways. Being able to view shapes in different ways opens doors for students who traditionally struggle seeing a shape in their head, and using it to solve their problems.

Sources

https://en.wikipedia.org/wiki/Pythagoras#In_mathematics

Engaging students: Defining the terms perpendicular and parallel

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 Diana Calderon. Her topic, from Geometry: defining the terms perpendicular and parallel.

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How has this topic appeared in high culture (art, classical music, theatre, etc.)?

– This topic of parallel and perpendicular appears in art in the early 1900’s, late 1910’-1930’s. The movement was widely known as De Stijl, which in Dutch means “the style”. This movement had characteristics of “abstract, pared-down aesthetic centered in basic visual elements such as geometric forms and primary colors.” , the two main artists of this artistic movement were Theo can Doesburg and Piet Mondrian. The artistic movement started because of a reaction to the end of World War I, “Partly a reaction against the decorative excesses of Art Deco, the reduced quality of De Stijl art was envisioned by its creators as a universal visual language appropriate to the modern era, a time of a new, spiritualized world order”. As seen below, there are multiple lines, all of which are either perpendicular to each other or parallel, “De Stijl artists espoused a visual language consisting of precisely rendered geometric forms – usually straight lines, squares, and rectangles–and primary colors.”.

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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.)

– When we think of geometry a lot of people instantly think of triangles, SOHCAHTOA, and other 2D or 3D shapes. But when I think of geometry I think of the Greeks and Euclid, the literal father of geometry, only because I learned about him in Dr. Cherry’s class. With that being said, Euclid was one of the first mathematicians to define the term parallel, in specific, parallel lines. In 300 BCE Euclid stated some definitions for the basics of geometry, then give five postulates, “The postulates (or axioms) are the assumptions used to define what we now call Euclidean geometry.” The fifth postulate is what we want to focus on because it is called the parallel postulate, “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.” He also states how to construct a perpendicular in Proposition 12, “To draw a straight line perpendicular to a given infinite straight line from a given point not on it.”, this construction states that by a given line AB and a point C not on the line then it is possible to construct a perpendicular on line AB.

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

– A good group project for the topic of parallel and perpendicular lines would be to allow the students to create a town. The requirements would be for the student’s town to be no bigger than 100 square inches, the students would have the liberty to create any quadrilateral shape as long as it meets the 100 square feet requirement. Another requirement that the project would have is that there must be at least 4 parallel streets, one perpendicular street that is only perpendicular for one of the parallel streets and finally one diagonal street that intersects 3 parallel streets. A town obviously needs to have shops so the students would be required to put shops within the town but must have an explanation as to why the shops were chosen. Finally the students must bring a physical final product, the shops must be in 3D form, the town area may be made with cardboard, cardstock or any material that would sustain the shops on top of it, the streets and corners of streets must be labeled with the corresponding angles. Finally, when they bring their final piece as a class we will walk around and allow the groups to present their product. As an exit ticket for presentation day the students must turn in the definitions of parallel and perpendicular in their owns words and how it was shown in their project product.

Citations:

o Mondrian Returns to France (Figure 1)
https://worldhistoryproject.org/1919/mondrian-returns-to-france

o De Stijl
https://www.theartstory.org/movement/de-stijl/

o The Three Geometries
https://mathstat.slu.edu/escher/index.php/The_Three_Geometries

o Euclid’s Elements I-XIII
https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html#posts