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

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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.

 

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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.

 

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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.

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