Component Form of Vectors

The class continues to develop some basic concepts of vectors. Students begin with a bell problem. Instead of just having defining the standard form of a vector and the component form of a vector, students determine how to find an equivalent vector whose initial point is at the origin.  This allows students to use their own problem solving techniques to find component form of a vector.

I let students discuss the questions before we share out. During the sharing time I ask students to explain their reasoning. I expect most students to use slope to find the vector. Most students draw diagrams to help solve the problem. I have students share their diagrams and explain how the diagram helped find the terminal point. The sharing allows students to communicate mathematically and helps struggling students see the reasoning of others.

From the bell problem students were able to put a vector in standard position and find the terminal point. I now share the terminology we use specifically, standard position and component form. Students read page 446 in Larson, "Precalculus with Limits, 2nd ed." This page has a lot of information to process. I ask the following questions to guide students to the key information from the page:

• What is meant when the book says "a vector is in standard position?"
• Why can a vector in standard position be identified by its terminal point?
• What is component form? (I depend on my physic students again. In physics they discuss the x component and the y component these students have used different notation than our book so I try to connect what they learned in physics with our notation. I ask the physic students, "what do you call these components in physics?")
• What is the notation for component form?
• If a vector is not in component form how can we find the component form? (With this question we discuss why we subtract the x values and the y values I will put a triangle on the board to make it easier to understand.
• How do we find the magnitude?

As the students answer these questions I go back to the bell problem. By referring to the bell problem students see that they converted to standard position without any instruction. We also write the component form of the bell problem.

Students work on a writing a vector in component form. When we work the problem in class I have students graph the original vector. When a student shares an answer I ask "How was this answer determined? So the first number is what component? What does that number tell us about the vector? What does the second number tell us about the vector? How does the component form of a vector help us draw equivalent vectors?"

To extend students thinking I ask: "What is the component form of vector QP?" This is an interesting discussion about how we are looking at both direction and distance. Students will begin by saying it is the same. Some students realize that we are going in a different direction but are not sure how to show that. After some conversation students say that the components will by negative (or the opposite of vector PQ).

For some students to have a good definition or rule for the key concepts we have discussed (magnitude, component form, direction) as well as notation.

To help students organize their ideas, I give students list of key ideas. Students then get into small groups (2-3). These groups determine the important information for each idea. As an example, I expect students to define magnitude as the distance between the beginning and end. I expect students to write out a formula on finding the magnitude or to give an example on how to find magnitude. We have not discussed the terms unit vector and zero vector when students ask me about these terms I ask "from everything we have learned so far how what might be meant by these terms?" Students are given about 5 minutes to discuss the ideas.

I now have  the class does a "Gallery Walk" to share their ideas. The terms have been put around the room on big pieces of paper. Each group goes to one term and writes out what they think is important about the term. After about a minute the groups move. When students move they add to the paper and may correct misconceptions that have been put on the page. We continue with the walk until every group gets back to their original term. This allows the groups to make sure they see everyone's ideas and to fix errors they might have.

As students move I review the information on the papers to make sure misconceptions are being corrected.

Students are given a problem to complete for an exit slip. I am able to assess which students are struggling with magnitude and writing a vector into component form with the exit slip.