We have been using the Law of Sines and Law of Cosines for the last few days to find parts of non-right triangles. Today we will be using the context of an airplane to see an application of these formulas while also giving students an introduction to vectors. I like this lesson because it gives students a good grasp of what a vector is and why they are used. Vectors can be very abstract, so it is helpful to start with something tangible to build their understanding.
This video clip from one of the great comedies of all time (in my humble opinion) might be a good intro to the lesson.
I begin the lesson by showing students slide #1 of this PowerPoint. I'll ask them if they notice anything strange about the travel itinerary and hopefully they will see that the return trip from Las Vegas to Detroit is about 30 minutes faster than the trip there. We will discuss why this is and relate it to winds and how they can make a trip faster or slower.
Next we move on to the second slide where I give a scenario of the airplane path that we will be investigating today. Before we start talking about vectors or doing any calculations, I want to see what their intuition will tell them about how winds can affect the path of an airplane. On slide #2 student will see an airplane's path and four different winds. I want them to discuss how each of the four winds will affect the speed and path of the plane. I will have them discuss for a few minutes with their table, and then we will talk as a class.
Next, students are given the task worksheet and are asked to find the "true course" of the airplane. I find that students usually need some clarification about what this means. Once that is clear, I want students to think about the new vector they should draw to represent the true course. From the Launch, my students are usually in agreement that the wind will cause the plane to speed up and that the angle will decrease; I use this intuitive knowledge as the springboard for the sketch of the resultant vector.
Many students think of connecting the two vectors to create the true course as shown by the red vector below.
Ask the class what they think about it and they will likely notice that the measure of angle B is greater than the measure of angle A, but we thought that it should be smaller. They will also likely notice that the length of the red vector is less than the length of the airplane's vector, but we said that the plane would speed up. It is good that they are thinking of triangles, but they will have to create a different triangle.
Students may not just think of vector addition on their own, so you may have to add in some direct instruction at this point. I tell students that the plane’s path and the wind’s path are examples of vectors. Next I talk about how vectors have direction and magnitude, unlike a scalar quantity. Giving examples of vector quantities and scalar quantities is very important to cement in this concept.
After looking at the first solution and determining that it is incorrect, I show students how to connect the vectors (tail to head) to create the resultant. Then I let them work and see if they can find the magnitude and direction of the new vector. Even when students understand what the resultant vector is going to look like, it is still challenging to find the appropriate tools to find the magnitude and direction.
After students have had time to work on finding the magnitude and direction of the true course vector, I will select a student who was on the right track to share their work on the document camera. There are a ton of concepts that can be used in the solution (linear pairs of angles, parallel lines, alternate interior angles, Law of Cosines, Law of Sines, etc.). I make sure that I use these vocabulary words so that students can communicate effectively about this solution strategy. Later in this unit we will use a more efficient strategy to find the magnitude and direction of the resultant vector, but for today I want to use the tools that we have used for the last few lessons.
After the initial Airplane Task has been wrapped up, we are going to generalize some information about vectors and talk about the nuts of bolts of them. This notes worksheet (and answer key) will give a good overview of vectors and let them explore vector operations. There are many different types of notations that you can use to represent vectors, so I like to use them interchangeably so that students get exposure to all of them.
I will lead a discussion about Questions 1 – 3 on the notes worksheet. Question #3 is important as it establishes the two different ways to describe vectors:
Students should be able to translate between the two representations fluently.
After this discussion of the first three questions, I plan for students to work independently to finish the worksheet. Question #7 will get students thinking about vector operations. I discuss this in the video below.
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Students may not completely figure out the vector operations today, but at least they have put some thought into it. Their conjectures will be the starting point for tomorrow's lesson and will help build a bridge to the mastery of these operations.