Engaging students: 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 Fidel Gonzales. His topic, from Precalculus: vectors in two dimensions.

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

When a student learns about vectors in two dimensions, they worry about the magnitude of the vector and the direction that it goes. The direction is kept within its limitations which are up, down, left, and right. A student might be curious as to how this topic can be extended further. The way it extends further is by extending vectors into higher dimensions. It is even possible to extend vectors to the sixth dimension! However, for the sake of showing how vectors in two dimensions extend to future courses in math, we will stick to three-dimensions. Learning about vectors in the second dimension creates groundwork to learn about vectors in the third dimension. With the third dimension, vectors could be seen from our point of view compared to seeing it in the two dimensions on paper. The new perspective of the third dimension in vectors includes up, down, left, right, forward, and backwards. Having the new dimension to account for will give students a bigger tie into how mathematics applies into the real world.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Vectors in the two dimension is used all around our everyday life and we as people rarely notice it. The most common use of vectors in our culture is a quantity displaying a magnitude and direction. This is normally done on a x and y graph. Now you might be asking yourself, I do not play any types of games that sound like this. I am here to tell you that you do. One game that iPhone users play without noticing this would be a game on gamepigeon called knockout. The game appears to be an innocent game of knocking out your friends’ penguins while keeping yours in the designated box. However, math is involved, and you probably didn’t notice. First you must anticipate where the enemy is going. Then you must decide how strong you want to launch your penguin troopers without making them fall out of the ring. Does that sound familiar? Having to apply a force (magnitude) and direction to a quantity. Congratulations, you have now had fun doing math. Next time you are playing a game, try to see if there is any involvement of vectors in two dimensions involved.

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

Vectors in two dimensions has many ways to be incorporated in the classroom. A way to do so while connecting to the real world would be having an activity where the students tell a robot where to go using vectors. The students will have a robot that can walk around and in need of directions. The students will be given maps and asked to create a path for the robot to end up in its destination. Essentially, programming the robot to navigate though a course solely using vectors. If the robot falls or walks too far, then the student will realize that either the magnitude was wrong or the direction. Some students might seem to think this would be impractical to the real world, however, there is always a way to show relevance to students. Towards the end of the activity, the students will be asked to guide me to around the class using vectors. Then to sweeten the deal, they will also be asked to show me on a map being projected to them how to get to McDonald’s. Students will realize that vectors in the second dimension could be used to give directions to somewhere and can be applied to everyday life. They will walk outside of the classroom seeing math in the real world from a different perspective.

References:

https://www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/v/visualizing-vectors-in-2-dimensions

Engaging students: Computing the cross product of two vectors

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 Chi Lin. Her topic, from Precalculus: computing the cross product of two vectors.

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

I found one of the real-life examples of the cross product of two vectors on a website called Quora. One person shares an example that when a door is opened or closed, the angular momentum it has is equal to $r \times p$ , where $p$ is the linear momentum of the free end of the door being opened or closed, and $r$ is the perpendicular distance from the hinges on which the door rotates and the free end of the door. This example gives me an idea to create an example about designing a room. I try to find an example that closes to my idea and I do find an example. Here is the project that I will design for my students. “If everyone here is a designer and belongs to the same team. The team has a project which is to design a house for a client. Your manager, Mr. Johnson provides a detail of the master room to you and he wants you to calculate the area of the master room to him by the end of the day. He will provide every detail of the master room in three-dimension design paper and send it to you in your email. In the email, he provides that the room ABCD with $\vec{AB} = \langle -2,2,5 \rangle$ and $\vec{AD} = \langle 5,6,3 \rangle$ . Find the area of the room (I will also draw the room (parallelogram ABCD) in three dimensions and show students).”

Reference:

https://www.quora.com/What-are-some-daily-life-examples-of-dot-and-cross-vector-products

https://www.nagwa.com/en/videos/903162413640/

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

This topic is talking about computing cross product of two vectors in three dimensions. First, students should have learned what a vector is. Second, students should know how to represent vectors and points in space and how to distinguish vectors and points. Notice that when students try to write the vector in space, they need to use the arrow. Next, since we are talking about how to distinguish the vectors and the points, here students should learn the notations of vectors and what each notation means. For example, $\vec{v} = 1{\bf i} + 2 {\bf j} + 3 {\bf k}$ . Notice that $1{\bf i} + 2 {\bf j} + 3 {\bf k}$ represents the vectors in three dimensions. After understanding the definition of the vectors, students are going to learn how to do the operation of vectors. They start with doing the addition and scalar multiplication, and magnitude. One more thing that students should learn before learning the cross product which is the dot product. However, students should understand and master how to do the vector operation before they learn the dot product since the dot product is not easy. Students should have learned these concepts and do practices to make sure they are familiar with the vector before they learn the cross products.

References:

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/x786f2022:vectors-and-matrices/a/vectors-and-notation-mvc

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/x786f2022:vectors-and-matrices/a/dot-products-mvc

How did people’s conception of this topic change over time?

Most people have the misconception that the cross product of two vectors is another vector. Also, the majority of calculus textbooks have the same misconception that the cross product of two vectors is just simply another vector. However, as time goes on, mathematicians and scientists can explain by starting from the perspective of dyadic instead of the traditional short‐sighted definition. Also, we can represent the multiplication of vectors by showing it in a geometrical picture to prove that encompasses both the dot and cross products in any number of dimensions in terms of orthogonal unit vector components. Also, by using the way that the limitation of such an entity to exactly a three‐dimensional space does not allow for one of the three metric motions (reflection in a mirror). We can understand that the intrinsic difference between true vectors and pseudo‐vectors.

Reference:

https://www.tandfonline.com/doi/abs/10.1080/0020739970280407

Engaging students: Vectors in two dimensions

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 Sarah McCall. Her topic, from Precalculus: vectors in two dimensions.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For such an applicable topic, I believe that it is beneficial to have students see how this might apply to their lives and to real world problems. I selected the following word problems because they are challenging, but I think it is necessary for students to be a little frustrated initially so that they are able to learn well and remember what they’ve learned.

1. A DC-10 jumbo jet maintains an airspeed of 550 mph in a southwesterly direction. The velocity of the jet stream is a constant 80 mph from the west. Find the actual speed and direction of the aircraft.

2. The pilot of an aircraft wishes to head directly east, but is faced with a wind speed of 40 mph from the northwest. If the pilot maintains an airspeed of 250 mph, what compass heading should be maintained? What is the actual speed of the aircraft?

3. A river has a constant current of 3 kph. At what angle to a boat dock should a motorboat, capable of maintaining a constant speed of 20 kph, be headed in order to reach a point directly opposite the dock? If the river is ½ a kilometer wide, how long will it take to cross?

Because these problems are difficult, students would be instructed to work together to complete them. This would alleviate some frustrations and “stuck” feelings by allowing them to ask for help. Ultimately, talking through what they are doing and successfully completing challenging problems will take students to a deeper level of involvement with their own learning.

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

I believe vectors are fairly easy to teach because there are so many real life applications of vectors. However, it can be difficult to get students initially engaged. For this activity, I would have students work in groups to complete a project inspired by Khan Academy’s videos on vector word problems. Students would split off into groups and watch each of the three videos on Khan Academy that have to do with applications of vectors in two dimensions. Using these videos as an example, students will be instructed to come up with a short presentation or video that teaches other students about vectors in two dimensions using real world applications and examples.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Immediately when I see vectors, I think of one specific movie quote from my late childhood that I’ll always remember. The villain named Vector from Despicable Me who “commits crimes with both direction AND magnitude” is a fellow math nerd and is therefore one of my favorite Disney villains of all time. So of course, I had to find the clip (linked below) because I think it is absolutely perfect for engaging students in a lesson about vectors as soon as they walk in the door, and it is memorable and educational. I would refer back to this video several times throughout the lesson and in future lessons because it is a catchy way to remember the two components to vectors. This would also be great to kick off a unit on scalars and vectors, because it would get kids laughing and therefore engaged, plus they will always remember the difference between a scalar and a vector (direction AND magnitude!).

References:

Engaging students: Vectors in two dimensions

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 Derek Skipworth. His topic, from Precalculus: vectors in two dimensions.

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

While it may be a cop-out to use this example since I am developing it for an actual lesson plan, I will go ahead and use it because I feel it is a strong activity. I am developing a series of 21 problems that will be the base for forming the students’ treasure maps. There will be three jobs: Cartographer, the map maker; Lie Detector, who checks for orthogonality; and Calculator, who will solve the vector problems. The 21 problems will be broken down into 7 per page, and the students will switch jobs after each page. The rule is that any vectors that are orthogonal with each other cannot be included in your map. There are three of these on each page, so each group should end up with a total of 12 vectors on their map. Once orthogonality is checked by the Lie Detector, the Calculator will do the expressed operations on the vector pairs to come up with the vector to be drawn. The map maker will then draw the vector, as well as the object the vector leads to. Each group will have their directions in different orders so that every group has their own unique map. The idea is for the students to realize (if they checked orthogonality correctly) that, even though every map is different, the sum of all vectors still leads you to the same place, regardless of order.

B. How does this topic extend what your students should have learned in previous courses?

Vectors build upon many topics from previous courses. For one, it teaches the student to use the Cartesian plane in a new way than they have done previously. Vectors can be expressed in terms of force in the $x$ and $y$ directions, which result in a representation very similar to an ordered pair. It gets expanded to teach the students that unlike an ordered pair, which represents a distinct point in space, a vector pair represents a specific force that can originate from any point on the Cartesian Plane.

Vectors also build on previous knowledge of triangles. When written as $\langle x,y \rangle$ , we can find the magnitude of the vector by using the Pythagorean Theorem. It gives them a working example of when this theorem can be applied on objects other than triangles. It also reinforces the students trigonometry skills since the direction of a vector can also be expressed using magnitude and angles.

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

The PhET website has one of the best tools I’ve seen for basic knowledge of two dimensional vector addition, located at http://phet.colorado.edu/en/simulation/vector-addition. This is a java-based program that lets you add multiple vectors (shown in red) in any direction or magnitude you want to get the sum of the vectors (shown in green). Also shown at the top of the program is the magnitude and angle of the vector, as well as its corresponding $x$ and $y$ values.

What’s great about this program is it puts the power in the student’s hands. They are not forced to draw multiple sets of vectors themselves. Instead, they can quickly throw them in the program and manipulate them without any hassle. This effectively allows the teacher to cover the topic quicker and more effectively due to the decreased amount of time needed to combine all vectors on a graph.