# 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.

# 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 again comes from my former student Shama Surani. Her topic, from Geometry: deriving the distance formula.

A1. What interesting word problems using this topic can your students do now?

By viewing examples on http://www.spacemath.nasa.gov, I came across the following word problem:

A beam of light, traveling at 300,000 km/sec is sent in a round trip between spacecraft located Earth (0,0), Mars (220, 59), Neptune (-3200, -3200), and back to Earth. If the coordinate units are in millions of kilometers, what are:

A)    The total round-trip distance (Earth, Mars, Neptune, Earth) in billions of kilometers?

B) The round trip time in hours?

I believe this problem is an interesting one to ask the students because I believe this question will pique the interests of the students especially if a video clip or visual is presented to grab their attention. This question allows me as a teacher to assess what the students know, and if they can apply the previous concepts learned to this new concept. By the end of the lesson, the students will be able to find out the total distance, and also apply previous concepts with distance = rate * time to figure out how many hours the round trip took.

By the end of the lesson, the students will be able to answer these questions. This problem builds on previous concepts taught so students can tie and see the connections among all topics.

http://www.nasa.gov/pdf/377674main_Black_Hole_Math.pdf

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

As a teacher, I can create an activity or project that involves the distance formula. I will provide a map of the United States, and have the students plan a trip across the USA covering at least 10 states, and making pit stops along the way of places they would want to visit, such as the Grand Canyon, Las Vegas, etc. The students will have to find the distance of the total trip, as well as the distance between each pit stop. This activity helps the students practice the distance formula while allowing the students to become familiar with the United States and interesting locations to visit in the United States. The students will know be able to see how the calculating distance is related to real life.

http://livelovelaughteach.wordpress.com/category/midpoint-formula/

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

Pythagoras, Euclid, and Descartes are the three main mathematicians who are most responsible for the development of the distance formula.  Pythagoras is acknowledged by many scholars as being the one to have invented the distance formula although much record in history has been lost during this period. He was born around 570 B.C. in Samos. As a Greek mathematician and philosopher, he traveled to other parts of the world to learn from other civilizations, and he always was seeking the meaning of life. Pythagoras was amazed with distances as he travelled to Egypt, Babylon, Arabia, Judea, India, and Phoenicia. He is the one credited for one of the first proofs of the Pythagorean theorem, a2 + b2 = c2. The distance formula is derived from the Pythagorean theorem.

Euclid, known as the father of Geometry, also contributed to the distance formula. His third axiom states, “It is possible to construct a circle with any point as its center and with a radius of any length.” If one considers the equation of a circle, x2 + y2 = r2, one will notice that the distance formula is a rearrangement of the equation of a circle formula.

Renee Descartes was the one who developed the coordinate system that allows connection from algebra to geometry. He took the concepts of Euclid and Pythagoras in order to relate the radius to the center point of the circle. Essentially, Descartes came up with the equations used for circles and distance between two points that are used today.

http://harvardcapstone.weebly.com/history2.html

References:

http://www.cs.unm.edu/~joel/NonEuclid/proof.html

http://harvardcapstone.weebly.com/history2.html

http://livelovelaughteach.wordpress.com/category/midpoint-formula/

# Engaging students: Finding the area of a circle

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 Rebekah Bennett. Her topic, from Geometry: finding the area of a circle.

Culture:

The area of a circle is used in our everyday life. Landscaping uses this topic quite a bit. Suppose a person wants to put a circular pool or even a fountain in their yard. The landscaper needs to know the area of the basic circle that is being used so that they can make sure there is enough land to build on. We also know contractors use this everyday too. When building a circular building, the contractor needs to know the area of the base of the building so that he/she can clear a big enough area. They also use this when building circular columns, such as the ones you would see on a big, fancy building. The contractor must know how much area the base (circle) takes up to see how much of the platform they have left to work with. Then he/she can now see how many evenly spaced columns will fit on the platform. A room designer also comes to mind. Let’s say if someone wanted a circular table placed in their living room, the designer needs to know how much space (area) the table takes up in order to figure out how much area is left in the room to fit other items comfortably. These are all instances where someone in the artistic world would need to use area of a circle.

Application and Technology:

To explore this topic, I would give each student a cut out of a circle, each circle having a different size. Then I would tell them to figure out the area of the circle. I would give them hints as to how would you use the radius, diameter, and circumference within a formula. I would suggest the idea of splitting the circle into even pieces, and then ask the students if there is a way that they can transform the pieces of a circle into a more familiar shape. The students would have about 5 minutes to experiment on their own and then I would show them this video.

This video shows the students a more in depth definition of area of a circle. The video actually derives the formula from the normal area formula of a parallelogram (base x height). Here we pull the whole circle apart, piece by piece to create a parallelogram. The video relates height to radius and base to ½ of the circumference. These are both previous terms that the student already knows. The guy in the video manipulates the area formula for a parallelogram to derive the area of a circle. This video is a great way to show students that there is more than one way of solving for the right answer but also more importantly, it shows where the formula for area of a circle actually comes from. This gives the student a justification as to how and why we created this formula, relating back to the exploration.

After watching the explanation from the video, the students would now have a chance to replicate the demonstration with their original circle. By having the students recreate the video demonstration themselves, it gives them a better understanding as to why the formula works like it does and they can see how the formula works with a guided hands on approach.

Curriculum:

From previous math courses, the student should already know the terms of a circle such as; radius, diameter and circumference. The student should know how to find the radius given the diameter, vice versa. The student knows that the circumference is the perimeter of a circle and how to find it, given the radius or diameter. They should already know the term area: space that an object takes up. The student should know how to find the area of a rectangle and parallelogram: (length x width) or (base x height). This activity shows how to relate the area of a circle to the area of a rectangle, given the radius and height, which is the same thing. The student can now create a formula for the area of a circle by using the same method as solving for area of a rectangle or parallelogram. The area of a circle extends the previous knowledge that every student should learn in algebra before entering a geometry class.

# Engaging students: Introducting translation, rotation, and reflection of figures

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 Isis Flores. Her topic, from Geometry: introducing translation, rotation, and reflection of figures.

B2: How does this topic extend what your students should have learned in previous courses?

In order for students to be able to be successful understanding, performing, and identifying translations, rotations and reflections there are a few things that they must have a grasp on from previous classes. Included in these topics is understanding the Cartesian plane and the different relationships between each quadrant. Knowledge of the plane will be extended when students began to work with different degrees of rotations around the plane. Students should also be able to perform several different tasks on the plane such as, plotting points and lines. Being able to perform such tasks will ease the transition of now working with more complex shapes on the plane. Since the topic deals with transformations of figures students must also have an understanding of the basic geometric figures and their different characteristics and classifications. Having a base knowledge of geometric shapes will aid the students when comparing different types of transformations. In previous courses students should also have acquired knowledge of the basic mathematical operations, (addition, subtraction, multiplication, division), which will enable them to perform specific dictated transformations better. The concept of basic mathematical operations will be extended to students as they explore how these operations may play out on a coordinate plane with geometric figures.

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

In any classroom there is always a variety of students with a variety of interests. One of these interests may include art, which can lend itself quite easily to the exploration of different transformations. A specific type of art which uses translations, rotations and reflections is called Geometric Abstraction. Geometric Abstraction became widely popular in the early 20th century making it an even closer connection for students. The art form uses different types of geometric shapes to create abstract and quite modern looking pieces of work. The fact that the art form is quite new compared to other forms of art does not prevent pieces from being high end items, and the monetary aspect may be another way to engage students. Showing students different pieces of art which were composed using geometric transformations and also showing how highly priced they are, is a great way to show the relevancy and demand for the topic.

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

As a teacher at times it is difficult to get students motivated and excited about a specific topic. A great way to give students motivation towards an activity is to give them a bit of autonomy. For translations, rotations and reflections a project that students may perform may be their own art work which would display their knowledge of the content. To even personalize the project even more students may be ask to include an object which is personal to them, for example if a student play soccer then a soccer ball would be an appropriate object for their art work. Students may be asked to also provide directions on their art work so that a classmate may replicate it. Perhaps to take a step further students may analyze each other’s art pieces and try to figure out what order of transformations created the finished piece. For students who may not feel as artistically inclined, or even as another class project, the option of going and finding a real life depiction of transformations may be offered. Students should provide evidence of their findings with an image. The task can be furthered challenged by asking students to find something in their school which depicts transformations. The first project will require students to show their proficiency in performing the transformation, while the second will call on them to show their understanding of what each transformation looks like.

References:

http://www.artspace.com/assume_vivid_astro_focus/starburst

http://www.artspace.com/magazine/art_101/art_101_geometric_abstraction

# Engaging students: Midpoint 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 again comes from my former student Christine Gines. Her topic, from Geometry: deriving the midpoint formula.

How has this topic appeared in pop culture?

Finding the midpoint between two points is a fairly common situation we find ourselves in daily. Take for example cutting a sandwich into two equal halves. Here you are estimating the midpoint between the ends of the sandwich. Maybe you want the bigger half of the sandwich though. In this case you first find the middle and then move slightly away. Whether we realize it or not, finding midpoints happens all around us and bringing this to students’ attention is crucial for their development of connections.

One way to aid these connections is to demonstrate how midpoints appear in our cultures. In particular, I found a popular music video “Meet Me Half Way” by The Black Eyes Peas. The video/song is about Fergie and Will.I.Am being apart and missing each other. Fergie’s solution is “Can you meet me halfway? Right at the borderline. That’s where I’m gonna wait… for you.” Fergie and Will.I.Am’s beginning locations are the endpoints in this scenario and they will meet at their midpoint. In the video, Fergie has already reached midpoint. Here, her lyrics are “Took my heart to the limit, and this is where I’ll stay. I can’t go any further than this.” This can be interpreted as a unique midpoint. If Fergie goes any further, she will no longer be at the midpoint. Her limit is the one midpoint. At the end of the video, Fergie and Will.I.Am are reunited at their midpoint.

After this connection is made, it could be reinforced by giving students specific coordinates of Fergie and Will.I.Am and asking students to find their midpoint. For example, Fergie and Will.I.Am were shown to be on different planets in the video. So, the teacher could give them the coordinates to Jupiter and the earth. If they succeed with this problem, a follow up could be to find the endpoint when you have Will.I.Am’s endpoint and their midpoint.

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

A common issue students face regarding formulas is memorizing them without fully comprehending the formulas. They say, “give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime.” So, let’s not just give students a formula, but teach them how to derive the formula by letting them explore the concepts for themselves. A good activity to let students do this is as follows:

In this activity students will Investigate finding the midpoint of a line segment and derive the formula for the midpoint of two points on a coordinate plane.

Have students work in groups of 3 or 4. Each group will have a sheet of large graph paper, markers, a ruler, dice and a penny.

Procedures:

1. Students will find two points by rolling dice and tossing penny (Dice represents number and penny represents positive or negative) and plot them.
2. They will draw a line to connect these two points.
3. Next, students can use the ruler to estimate where the midpoint should be.
4. Have students investigate ways to accurately find the midpoint of the segment and challenge them to find a formula as well.

Students can create several graphs so that they can recognize the patterns. By letting them draw and plot their own graph, students will more readily realize that the midpoint is exactly in between the two x-values and the two y-values. This will then hopefully lead students to recall how to find the average of two numbers, which is essentially what the formula is. It is important that students make this connection to their previous knowledge and to guide students through this exploration, teacher can ask leading questions such as:

• What could you use to represent the numbers so you can write a formula?
• How did you find that midpoint?
• Are you sure that is really the midpoint?
• How can you find the number in between two different numbers?

I don’t know about you but I’ve always thought the best educational games are the ones that actually feels like a game and not just something your teacher is making you do. This is exactly how the game “Entrapment” by The Problem Site feels like. Entrapment is actually a puzzle game. The object of the game is to create line segments such that all the given dots are midpoints to these segments.

More specifically, every red dot must be the midpoint of a line segment connecting two gray dots on the playing field. In the image above, the player is one move away from finishing since there remains one red circle which is not a midpoint. This puzzle is not only addicting, but it teaches students to recognize the relationship of x and y (individually) to the midpoint. After completely only a few of these puzzles, this relationship becomes part of your strategy, which in turn pushes students further away from memorization and brings them closer to comprehension. This puzzle brings all these educational benefits, yet it just feels like you’re playing a game!

http://www.theproblemsite.com/games/entrapment.asp#.UxF5ImJdXHQ

# 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 again comes from my former student Allison Metlzer. Her topic, from Geometry: defining the terms perpendicular and parallel.

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

The concepts of perpendicular and parallel will be implemented in many of my students’ future mathematics courses not only in high school, but also in college. In algebra, the students are asked to find the slope or the rate of change. In looking at the slope, students are asked to find if it’s parallel or perpendicular to another function’s slope.

In geometry, many shapes have properties that define them as having parallel or perpendicular sides (i.e. squares, rectangles, parallelograms, etc.). Also, in order to decide if triangles are similar, their corresponding sides must be parallel. In order to use the Pythagorean Theorem, the triangle must be right angled or have the two legs perpendicular to one another.

In calculus, students are asked to find orthogonal vectors which are also defined as perpendicular vectors. Also, calculus incorporates concepts from algebra and geometry which in turn, include parallel and perpendicular lines.

Therefore, many, if not all of my students’ future math courses will use the topics parallel and perpendicular. Thus, it would be important for me to teach them the two concepts correctly now so that there wouldn’t be any misconceptions in the future.

C3. How has this topic appeared in the news?

One big thing the news talks about every two years is the Olympics. Using the concept of parallel and perpendicular, the constructions are made for all of the different events. Apparent examples of events incorporating parallel lines are track, speed skating, and swimming. The one I will focus on is swimming, namely because it is a very popular Olympic event and one of my favorites. Pictured below is an Olympic swimming pool of 8 lanes. Do the lanes appear to be parallel? Two things that are parallel are defined as never intersecting while also being continuously equidistant apart. One can clearly see the lanes of the pool never intersect. If they did, then the contestants could interfere with one another. Also, because the Olympics is a fair competition, the lanes are equidistant in order to give each contestant a fair and equal amount of room.

Because the Olympics is a well-known event featured in newspapers, articles, and on TV, the students will be able to understand this real world application of parallel and perpendicular.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Before I would play the video, https://www.youtube.com/watch?v=vnnwfcDcNlY, I would first ask the students to think of as many examples they can of parallel and perpendicular in the real world. After about a couple of minutes, I would tell them to keep those in mind and see if the video included any they didn’t think of. I would play the video from 1:25 to 3:05 which is the portion that displays all of the examples. It has clear pictures of recognizable objects which incorporate parallel or perpendicular lines. Also, the video has labels on the pictures to even more clearly describe where the components of parallel and perpendicular lines are. I believe that the initial brainstorm along with this video would get the students thinking about the importance of parallel and perpendicular lines. Also, I would make the connection that those examples would not be considered parallel or perpendicular unless they met the following definitions. Then I could explicitly define both parallel and perpendicular.

Thinking of real world examples, and seeing pictures of them will help the students understand what parallel and perpendicular lines should look like. After they have this initial understanding, they then could get a better grasp of the definitions. Also, they would recognize the importance of following the definitions to correctly construct objects involving parallel and perpendicular lines.

References:

Detwiler, dir. Intro to Parallel and Perpendicular Line. YouTube, 2010. Web. <https://www.youtube.com/watch?v=vnnwfcDcNlY >.