# 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 >.