Engaging students: Using vectors in two dimensions

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 Zacquiri Rutledge. His topic, from Precalculus: using vectors in two dimensions.

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Add vector A and vector B, what do you get? How about when we take the dot product of A and B? What is the magnitude of A, B, and A+B? All of these are basic questions that a teacher might ask their students during a basic high school pre-calculus class. However, how does the teacher respond when a student asks “Where am I ever going to see this again”? In mathematics, a student might never see vectors again unless they take higher math such as Calculus I through III, or possibly Linear Algebra. During the first two courses of Calculus students will continue to expand on the ideas of two dimensional vectors by talking about the path an object might take through the air after leaving a cannon or being thrown off a cliff. Calculus III (or vector calculus) however is a much stronger example of how vectors will be used in further education of mathematics. During this class students will not only look at two dimensional vectors and review simpler ideas, but they will expand these ideas into the three dimensional world creating three dimensional vectors. Here students will discuss what kind of shape or planes a combination of three vectors might create.

A scientific use for two dimensional vectors is in physics. During a physics class, students talk about forces that act on objects as they move or when an object hits another. To do this, students draw vectors to represent the magnitude of the force that is acting on the object and the direction the force pushes or pulls the object. For example, in the previous paragraph it was mentioned about an object being shot from a cannon and the students measuring the path the object might take. In physics, the students might do the exact same thing, but by looking more in depth at the forces acting on the object. Forces might include the force of the cannon firing the object at a certain angle into the air, gravity pulling that object toward the ground, and even the friction of the air on the object as it soars through the air. Each one of these forces is acting on the object as it moves, either helping the object move farther and faster or attempting to slow it down. However, two dimensional physics is not the end of vectors; just like calculus, physics goes on to discuss what happens to objects in a three dimensional world and the forces that act on them. So a very easy answer to give the student asking where he/she will see vectors again is in every day real life.

 

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You have explained to your students that vectors are in everyday life and they still do not believe you. You have shown them countless examples on the board, drawing pictures of airplanes and the paths they fly through the air, objects being dropped from a cliff, objects being shot from a cannon, et cetera, and they still do not believe that vectors have any importance or use! Then you simply ask, has anyone ever seen a show called MythBusters on the Discovery Channel? Now MythBusters is a very well-known show, not only because it has been around for twelve years, but also for some of the crazy things that they test in the name of science. For example, some of my personal favorites include them making a boat out of pykrete, the many episodes on the uses of duct tape, and testing if a bullet dropped at the same time as a bullet shot from a gun will hit the ground at the same time. The great thing about this show is it is full of great examples of how physics affects things in real life. Also, not only do they test the myths, they explain how they are testing them, why they are testing them the way they are, and why it makes sense scientifically or does not. For example, during the bullet episode, they explain that once the bullet is shot from the gun, the only forces acting on the bullet are gravity and air friction. The only forces that would be acting on the bullet dropped would gravity and air friction as well. So in theory these two bullets should hit the ground at the exact same time if they are projected from the same height. By the end of the episode they had proven this by figuring out the best way to set up a live test and using a high speed camera to measure the time it took for each to hit the ground. For a high school class, this would be very easy to draw on a chalk board and walk the students through the thought process of why this happens using vectors to draw out the forces.

 

 

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Finally, the Internet gives us access to a lot of videos. This would allow a teacher who is talking about Mythbusters and their amazing examples of vectors in motion the chance to display quick clips of some of their tests. Of course the teacher will need to have researched a few before class in order to make sure they can be used as vector examples, but after a video has been played the teacher could ask the students to explain why the test was either plausible or false. On a small scale this video, https://www.youtube.com/watch?v=BLuI118nhzc , works great to show how a truck moving at the same speed as a soccer ball being shot from the back cancels the two forces, leaving gravity as the only force acting on the ball. Using vectors, a teacher could explain how one vector is positive and the other is negative of the same magnitude, cancelling the other out. Then show how only one vector on the ball remains, pulling the ball in that direction.

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