# 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. 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.”.  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. 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 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

# Engaging students: Parallel and perpendicular lines

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 Algebra: parallel and perpendicular lines. A2. How could you as a teacher create an activity or project that involves your topic?

An activity can be done with students by giving them a map, with a series of roads that run perpendicular or parallel to each other, asking them to identify pairs of perpendicular and parallel roads. To go beyond this, students can then find the slopes of a set of perpendicular or parallel lines on their own, then be asked to identify how they relate to one another. This will eventually lead them to being able to come up with a general rule to finding lines that are perpendicular or parallel to each other. Students can then be asked to create their own streets that will be perpendicular or parallel to some of the streets given. After this, students should be confident going from the representational model of perpendicular and parallel lines to graphing them on a cartesian plane. B2. How does this topic extend what your students should have learned in previous courses?

Studying perpendicular and parallel lines builds on a student’s knowledge of being able to calculate equations of lines and slopes given different amounts of initial information. It extends their knowledge of calculating slopes, and allows them to do it in reverse. Instead of getting two points to find the slope of the line, they may be given one point and the equation of a perpendicular or parallel line. This allows students to extend and apply their knowledge of linear equations, and gives them more situations to apply it to. This can then be extended to more challenging word problems, challenging students to come up with issues that require related slopes. E1. How can technology be used to effectively engage students with this topic?

Desmos can be very useful with engaging students in anything related to geometry or graphs. There are many resources within the website beyond just graphing two lines and viewing the relationship. A teacher can create their own activities within the website to allow students to explore a concept such as perpendicular and parallel lines, or they could use a pre-existing one created and shared by another educator. These activities give a great visual model of how perpendicular and parallel lines look, and then allow it for students to easily get the equations for each of the lines. Using Desmos can give students the capabilities of generating formulas and relationships on their own, without needing to be told what they are from their teachers. This will allow students a quicker path to mastery of the topic, and will lead them to applying it in a wider variety of areas more quickly than a student who is just told that slopes of parallel lines are equal and slopes of perpendicular lines are opposite reciprocals.

# 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 Tinashe Meki. His topic, from Geometry: deriving the term midpoint. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

During political elections, we usually hear how candidates are projected to do as the election moves forward. An important marker that usually separates likely candidates to win is the midpoint. Different new channels and news castor tend to use the phrase “midpoint of the election…”, or “midway through the election…” as ways to signify a halfway marker in time or events. The use of midpoint in news is used to describe halfway mark of time, events, distance etc. It’s a flexible word which gives its viewers a marker of how they can predict future events, time or distance. The uses of midpoint is inherently powerful because it simplifies and organizes ideas for views. For example, during time election there are so many stories being reported, different polls and various interpretation of how candidates are doing. Once the midpoint of the elections is reached, news anchors and new outlets provide the viewers with a consensus on how the election is going. That information is better received by the viewers because they can organize all the information they have received and create the own opinions for the second half of the election. How could you as a teacher create an activity or project that involves your topic?  What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

https://mathcs.clarku.edu/~djoyce/elements/bookIII/bookIII.html

This topic allows the teacher to simultaneously teach students about mathematical history and provide an engaging activity. I think the best way introduce students to the definition of a midpoint would be to have the students find the midpoint themselves, describe what they have found in their own words then provide them with a formal definition. A way to do that would be to show students how to bisect a line using Euclidian tool (ruler and compass) as the ruler, then have the students name the point where the line is bisected. Ask students to describe that point in their own words about the line. This activity would allow the instructor to introduce students to Euclidean geometry. The cool thing about using Euclidean geometry is that it allows students to visualize geometric concepts. It would provide them concrete understating of geometric topics. How have different cultures throughout time used this topic in their society?

https://www.learner.org/courses/learningmath/geometry/session1/part_c/index.html

https://www.ics.uci.edu/~eppstein/junkyard/origami.html

https://plus.maths.org/content/power-origami

An interesting approach to define midpoint would be to use origami geometry. Much like Euclid constructions, Origami offers similar constructions and definitions for geometry terms. Origami is Japanese art form that has been around since 200.AD. “Modern mathematicians Humiaki Huzita and Koshiro Hatori devised a complete set of axioms to describe origami geometry — the Huzita–Hatori axioms.” Among these axioms, one of them defines and constructs a proof for the midpoint. Having students construct the midpoint using Huzita and Hatori would be an interesting way to not only introduce the definition of midpoint, but also provide a different approach of explaining geometric concepts.

# Engaging students: Deriving the distance formula

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 Peter Buhler. His topic, from Geometry: deriving the distance formula. How could you as a teacher create an activity or project that involves your topic?

Although the distance formula may be introduced as part of the Geometry curriculum, it also has applications in Algebra and even Pre-calculus. This allows for many possible applications, as it can be used in various ways. One project that students could be assigned to is by modeling something in real life on a coordinate grid, and using the distance formula to calculate various distances within that real life object or place. An example of this could be to take a baseball diamond and use the fact that the bases are 90 feet apart, and calculate the distance between the corners on opposite sides. Another example could be to overlay a map of their town onto a coordinate grid and measure the distance between places that they usually visit. These students can fact check the distances by plugging them in to Google Maps. One aspect of this project to be careful of is to make sure that students are using the distance formula, and not the Pythagorean Theorem. Allowing the students to present their findings could spark curiosity into how mathematics is used in everyday life by city planners, architects, engineers, and in other careers. How has this topic appeared in high culture?

The following piece of artwork was created by Mel Bochner and titled, Meditation on the Theorem of Pythagoras. While immediately this picture appears to be related to the proof of the Pythagorean Theorem, There are also applications to the distance formula. This artwork could be a great engaging activity for students as they come into class, simply by reflecting on what can be seen. A challenging question would be to ask students to guess how many hazelnuts they think the artist used to create this artwork (without counting each piece). It should be noted that each corner of the triangle consists of two corners of the squares, so the answer is not simply 9+16+25, but you must subtract off how many are shared.
We can apply this to the distance formula by asking students how to relate the Pythagorean Theorem with the distance formula. Having students compare and contrast these two mathematical equations could provide excellent discussion. As an instructor, you can also overlay this artwork onto a coordinate grid and have students use the distance formula to calculate the various side lengths and confirm that it works. The three mathematics who are primarily responsible for what we know as the distance formula are: Euclid, Pythagoras, and Descartes. Euclid stated in his third Axiom that “it is possible to construct a circle with any point as its center and with a radius of any length”. This matters because the distance formula is a corollary of the circle formula. Pythagoras then took this idea, and proceeded to invent the Pythagorean Theorem, which can be easily converted to the distance formula. Later on, Descartes applied this to the coordinate system, in an event consisting of the union of algebra and geometry.
While this material may seem fairly dry to middle school or high school students who are first learning the Pythagorean Theorem, there are certainly some applications that can make the history more appealing. One such application is to ask the students to connect the formula of a circle with the distance formula, and discuss how they are related. This would provide excellent discussion about how Euclid and Pythagoras may have begun their study of the distance formula. Another application could include assigning students to study one of these three mathematicians, and having them provide several interesting facts about the person they chose to study. Consequently, when introducing the distance formula, students will be familiar with those who had a huge impact on the development of the distance formula.

# 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 Danielle Pope. Her topic, from Geometry: deriving the term midpoint. How could you as a teacher create an activity or project that involves your topic?

Introducing the definition of a midpoint in the classroom will take using class time to let students explore for themselves. The activity that I would make my students do is have the entire class stand up and have 2 students stand at opposite sides of the room. I would then ask my students to line up shoulder to shoulder. Once they were in a straight line I would ask “who is perfectly in the middle of this line?” This is where I would give my students 10 minutes initially to come up with various ways of how they would prove a student was in the middle of the line. Various “proofs” that they could tell me would be that there is exactly the same number of people on each side of the middle person. If that answer was given I would make an odd number of students stand in line and ask the same question of “Who is in the middle”? They would have to reconsider this answer because they couldn’t cut the student in half but I would hope that they would come to the conclusion that they would have to half the person in order to find the perfect center. Another “proof” that they may give me is measuring the distance from one end to the other and half that distance to find the person in the middle. This can also start that same conversation of how we would find the exact “midpoint” without cutting the person into pieces. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

To get just a basic definition of the midpoint, we can look at the lingo used in all sporting events. All sports have some form of a season that lasts for a certain amount of time. For this example specifically, I will be looking at the football season. Towards the middle of the season teams will know what to expect by the end. Most of the stats and predictions for teams are made already by the middle or midpoint of a season. In this article about football it relates to what changes various teams needed to make by the middle of their season. Just in the article itself, it says that “we’re now at the midpoint of the NFL season, and while some things are beginning to take shape, there’s still plenty of football left to be played.” In this context, students can understand that midpoint is being used to describe the middle of a football season. With this knowledge, they can use those context clues and just add the numbers given to them. One of the most important people in mathematics to date would have to be Euclid. Euclid’s book, The Elements, is still the backbone of all mathematics taught from kindergarten to college. One artist took this book or manual to mathematics and put it in the form of artwork. Crockett Johnson is an artist who bases his work off of mathematics. He takes the complicated proofs, lemmas, and theorem that have been proved and puts those in a form that we see as beautiful. One piece that uses mostly all midpoints titled “Bouquet of Triangle Theorems”. This piece is based off of the many of Euclid’s propositions about triangle just used together in one piece of art. For example “the midpoints of the sides of the large triangle in the painting are joined to form a smaller one.” Giving students a copy of this picture they can find various characteristics given a ruler and other tools that can help them possibly come to this conclusion that Euclid already proved. Crockett’s pieces can also be seen at the Smithsonian so that could show kids that math really does show up everywhere in our world even in unexpected places. http://www.foxsports.com/nfl/gallery/every-nfl-teams-biggest-weakness-at-the-midpoint-of-the-2016-season-110116

# Engaging students: Parallel and perpendicular lines

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 Jacobs. His topic, from Algebra: parallel and perpendicular lines. D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid is one of the most famous mathematicians of all time. His fame rests mostly on his 13 books commonly referred to as Euclid’s Elements. Euclid’s Elements are said to have a greater impact on the human mind that any other book except for the bible. Euclid contributed to the development of this topic based off the fact that his Elements have been used for centuries for teaching foundational geometry. The importance of Euclid’s books come from the minimal assumptions made, and the natural progression from simple results to more complex results. Euclid starts of listing 23 definitions and 5 postulates in which uses to prove theorems. His books contain over 400 theorems and proofs which layout the guidelines for how we use geometry today. E1. How can technology be used to effectively engage students with this topic?

Desmos.com is a great website website that allows you to pick out activities your students can do. They have some activities regarding parallel and perpendicular lines where students shift the lines to make them parallel or perpendicular. I have used this website before regarding parabolas and students are fully engaged. Desmos has plenty of activities to choose from to find the right fit for your class, so do not be afraid to look around for a while. You can sign in as a teacher and make a code for your students to get into the activity. There are even some word problems so you can get a better understanding of what your students are thinking. I think Desmos is best used at the end of a topic, more as a general review over everything because the activities go through topics pretty fast. B1. How can this topic be used in your students’ future courses in mathematics or science?

Students will continue to use parallel and perpendicular equations throughout their mathematical career. I am now in vector calculus and I am still using parallel and perpendicular lines in 3-dimensional planes. With that being said parallel and perpendicular lines are not going to disappear as you go further into math, in fact you have to start using different methods to find the parallel and perpendicular lines the farther you go. Soon it will no longer be as simple as duplicating the slope or finding the reciprocal. Parallel and perpendicular lines also play a key part in physics regarding vectors just as they do in vector calculus, when you try to find equilibrium forces.

# 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 Christine Gines. Her topic, from Geometry: deriving the term midpoint. Euclid was an Alexandrian Greek mathematician who created Euclidean geometry, and is also known as the Father of Geometry. He created a book called Elements, becoming one of the most influential works in mathematics’ history. Not much biographical information is known about him so many researchers believe he was not just one man, but rather a fictional character created by a team of mathematicians. This hypothesis however, is not well accepted by todays scholars. Euclid’s book Elements consists of 13 separate books, all bounded together, which is now what many high school math courses are based off of, – especially geometry. In book one proposition 10, the bisection of finite straight line is constructed and proved, which is also the construction of the midpoint of a finite segment. Many of the books works and theories have been taken, molded and manipulated throughout the years by mathematicians in order to form new and innovative ideas and theories. For example, being able to construct a mid point by using only circles. Mathematicians have challenged Euclid and his proofs many times, thus leading to great discoveries and theories, such as the discovery of doing his constructions in less steps (par value) and other types of math, but they still haven’t disproven much.

http://blog.yovisto.com/euclid-the-father-of-geometry/

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

http://math.stackexchange.com/questions/227285/constructing-the-midpoint-of-a-segment-by-compass Creating a midpoint hands on before seeing a precise definition is a great strategic way for a student to end up with reasonable definition of sed midpoint. According to Euclid, knowing how to create a midpoint with a ruler and compass can lead to the capability of creating other common shapes like circles, triangles, and squares. Common shapes are all around us in each and every material thing, but not many people think like a mathematician does. For example, a mathematician thinks the roof of a house looks like a triangle and not just an every day roof, a hot tub looks like a circle, a door looks like a rectangle and an infinite number of more examples. There is also more in depth use of common shapes like these. Films create their characters according to the correlation of shapes and emotions. For example, a villain is created to cause terror, fear, and intimidation; the type of shapes that portray those emotions are sharp and jagged, a lot like triangles are. The video attached does a great job on putting together a series of popular films and demonstrating how common shapes on characters and scenes manipulate the viewer’s feelings. This will allow the students to see how being able to define a midpoint leads to the creation of other shapes, and also their role in pop culture and how much it impacts them without even noticing. Defining the midpoint is not only limited to a finite line segment. In algebra two the students will learn and have to find the vertex of a parabola. Finding the midpoint of a quadratic equation is equivalent to finding the vertex, because the value x is the axis of symmetry of the parabola. Being able to derive the axis of symmetry is also a beginning step to writing an equation in vertex form and completing the square. The comprehension of the midpoint formulas, axis of symmetry, and vertex form will form a direct path to the introduction of conics and deriving formulas for them. In addition, students are also taught about area approximation under a curve and how to calculate it. When students are first being introduced to the topic they are taught a technique called Riemann sum. Riemann summation is best approached with partitions of equal size over an interval. There are four methods to calculate such technique left Riemann Sum, Right Riemann Sum, Trapezoidal Rule, and Middle Sum. To calculate Middle Sum method, the student will have to approximate the function at the midpoint of partitions.

# 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 Irene Ogeto. Her topic, from Geometry: defining the terms perpendicular and parallel. How could you as a teacher create an activity or project that involves your topic?

In order to explore the terms perpendicular and parallel the students could create their own parallel and perpendicular lines using a compass and ruler. I would provide compasses and rulers for the class and we would do the activity together. I would walk the students through the step-by-step process. This activity would allow the students to not only see parallel and perpendicular lines but to actually create them. We could explore different methods of constructing parallel lines about a given point: Angle copy method, translated triangle method, rhombus method. Likewise, we could explore different methods of constructing perpendicular lines: perpendicular from a line through a point, perpendicular from a line to a point and perpendicular at the endpoint of a ray. If we have time we could also go in depth and prove why these constructions work. In addition, the students can use Geometers Sketchpad to do the constructions as well.   How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The topic of parallel and perpendicular lines has appeared in the “real” portion of the Cyberchase show on television. In this episode, Harry is meeting his cousin to get tickets to go to a game. Harry and his cousin are both on the same street but have trouble meeting up. Harry decides it would be best to meet his cousin where Amsterdam Ave intersects with 79th street. This video could be shown at the beginning of a lesson as an engage when defining the terms parallel and perpendicular. Parallel and perpendicular lines are commonly found in roads and streets. Although this does not show that Amsterdam Ave and 79th street necessarily intersect at a right angle, it shows the difference between parallel and intersecting lines.

http://pbskids.org/video/?guid=302989e5-9265-4110-ac81-0b1e89ac2c40 How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Parallel and perpendicular lines are all around us, specifically in high culture. Parallel and perpendicular lines can be found in architecture. Many buildings have features that contain parallel and perpendicular lines. Most windows have parallel and perpendicular lines. Skyscrapers such as the New York Times Building, churches, schools, hospitals are all examples of some buildings that contain parallel and perpendicular lines. Parallel and perpendicular lines are also found in knitting, crocheting, and quilting patterns. Crochet scarfs can be made with parallel line patterns. Quilting is a technique which requires attention to detail and knowing the terms parallel and perpendicular can help speed up the quilting process. In addition, parallel and perpendicular lines can be found in art paintings. There are many paintings in the Dallas Museum of Art that contain parallel and perpendicular lines. An example is the painting Ocean Park No.29 done by American painter Richard Diebenkorn (1922-1993).

References:

http://www.mathopenref.com/constperpendray.html

http://www.pbslearningmedia.org/resource/6fb2456e-3696-4daa-863e-f76ea17f8f61/6fb2456e-3696-4daa-863e-f76ea17f8f61/

http://pbskids.org/video/?guid=302989e5-9265-4110-ac81-0b1e89ac2c40

https://www.dma.org/collection/artwork/richard-diebenkorn/ocean-park-no-29

# Engaging students: Parallel and perpendicular lines

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 Emma Sivado. Her topic, from Algebra: parallel and perpendicular lines. D.1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

I would take my students back to the time of Euclid of Alexandria, around 300 B.C., and his great book The Elements. Little is known about Euclid except the book he left behind which is the foundation of geometry, algebra, and number theory, still to this day. Euclid wrote this book in an axiomatic way, this means that he assumes common notions, definitions, and postulates to be true and then bases all his propositions and axioms on these assumptions. Does this sound like the way that we do mathematics today? To understand how influential and enduring the Elements is I would present this incredible fact; other than the Bible, Euclid’s Elements is the most published, translated, and studied of all books in the world.

Now we would put on our Euclid caps and turn to Proposition 12 and Proposition 31. These propositions tell us how to draw parallel and perpendicular lines based only on the definitions, common notions, and axioms of Euclid. We would do the constructions step by step, straight out of Euclid’s Elements.

http://www.britannica.com/biography/Euclid-Greek-mathematician

http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html A.2: How could you as a teacher create an activity or project that involves your topic?

To engage the students in the lesson on parallel and perpendicular lines, instead of sitting in class and listing real world examples of parallel and perpendicular lines, I would take the students out of the classroom and take a tour through the school like a bird watching group except our goal is to list all the parallel and perpendicular lines inside and around the school. We could go to the cafeteria, the gym, and walk around the outside of the building. When we got back to class we could create a long list of all the parallel and perpendicular lines that we see to hang on the wall during this unit. After we list the examples, I could ask some thought provoking questions:

“Why are these parallel and perpendicular lines important?”
“How would the world be different without parallel and perpendicular lines?” E.1: How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

A great activity I found on parallel and perpendicular lines involves using a graphing calculator to discover the similarities in slope between parallel and perpendicular lines. First, you give the students a list of equations to graph on their calculator. Next, you ask them to compare the graphs and identify which lines are parallel and which are perpendicular. Last, you ask them to compare the slopes of the parallel and perpendicular lines. Hopefully, they will discover that parallel lines have the same slope and perpendicular lines have the opposite reciprocal slope. This activity can be done easily because the students should already be familiar with graphing calculators, slope, and y-intercept. The activity would not take much time and can easily be differentiated based on the skill level of the students in your class. You can give some students difficult numbers or more lines to analyze if they finish the initial activity quickly. Also, you could take this one step further and give the students large sheets of graph paper and let them draw the lines and present their findings in front of the class.

# Inverse Functions: Arctangent and Angle Between Two Lines (Part 25)

The smallest angle between the non-perpendicular lines $y = m_1 x + b_1$ and $y = m_1 x + b_2$ can be found using the formula $\theta = \displaystyle \tan^{-1} \left( \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \right)$.

A generation ago, this formula used to be taught in a typical Precalculus class (or, as it was called back then, analytical geometry). However, I find that analytic geometry has fallen out of favor in modern Precalculus courses.

Why does this formula work? Consider the graphs of $y = m_1 x$ and $y = m_1 x + b_1$, and let’s measure the angle that the line makes with the positive $x-$axis. The lines $y = m_1 x + b_1$ and $y = m_1 x$ are parallel, and the $x-$axis is a transversal intersecting these two parallel lines. Therefore, the angles that both lines make with the positive $x-$axis are congruent. In other words, the $+ b_1$ is entirely superfluous to finding the angle $\theta_1$. The important thing that matters is the slope of the line, not where the line intersects the $y-$axis.

The point $(1, m_1)$ lies on the line $y = m_1 x$, which also passes through the origin. By definition of tangent, $\tan \theta_1$ can be found by dividing the $y-$ and $x-$coordinates: $\tan \theta_1 = \displaystyle \frac{m}{1} = m_1$.  We now turn to the problem of finding the angle between two lines. As noted above, the $y-$intercepts do not matter, and so we only need to find the smallest angle between the lines $y = m_1 x$ and $y = m_2 x$.

The angle $\theta$ will either be equal to $\theta_1 - \theta_2$ or $\theta_2 - \theta_1$, depending on the values of $m_1$ and $m_2$. Let’s now compute both $\tan (\theta_1 - \theta_2)$ and $\tan (\theta_2 - \theta_1)$ using the formula for the difference of two angles: $\tan (\theta_1 - \theta_2) = \displaystyle \frac{\tan \theta_1 - \tan \theta_2}{1 + \tan \theta_1 \tan \theta_2}$ $\tan (\theta_2 - \theta_1) = \displaystyle \frac{\tan \theta_2 - \tan \theta_1}{1 + \tan \theta_2 \tan \theta_1}$

Since the smallest angle $\theta$ must lie between $0$ and $\pi/2$, the value of $\tan \theta$ must be positive (or undefined if $\theta = \pi/2$… for now, we’ll ignore this special case). Therefore, whichever of the above two lines holds, it must be that $\tan \theta = \displaystyle \left| \frac{\tan \theta_1 - \tan \theta_2}{1 + \tan \theta_1 \tan \theta_2} \right|$

We now use the fact that $m_1 = \tan \theta_1$ and $m_2 = \tan \theta_2$: $\tan \theta = \displaystyle \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$ $\theta = \tan^{-1} \left( \displaystyle \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \right)$ The above formula only applies to non-perpendicular lines. However, the perpendicular case may be remembered as almost a special case of the above formula. After all, $\tan \theta$ is undefined at $\theta = \pi/2 = 90^\circ$, and the right hand side is also undefined if $1 + m_1 m_2 = 0$. This matches the theorem that the two lines are perpendicular if and only if $m_1 m_2 = -1$, or that the slopes of the two lines are negative reciprocals.