##
* *Reflection: Developing a Conceptual Understanding
Airplane! - An Introduction to Vectors - Section 4: Summarize and Extend

I really love this task because it gives us a basis to always return to when we get stuck with vector problems. From now on, whenever we have confusion about vectors they will always just be airplanes!

For example, if a student is confused about what a unit vector is, we can just think of it as a very slow airplane that is going 1 mile per hour. If a students doesn't understand resultant vectors, we can always refer back to it being the true course of an airplane with a wind acting on it.

I try to front load real-world contexts as much as I can because it always gives us something to refer back to when we reach a conceptual hurdle.

*Vectors will Always be Airplanes*

*Developing a Conceptual Understanding: Vectors will Always be Airplanes*

# Airplane! - An Introduction to Vectors

Lesson 4 of 12

## Objective: SWBAT find the resultant vector as the sum of two vectors.

#### Launch

*10 min*

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.

#### Resources

*expand content*

#### Explore

*15 min*

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.

*expand content*

#### Share

*10 min*

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.

#### Resources

*expand content*

#### Summarize and Extend

*15 min*

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:

- As a magnitude and direction
- As horizontal and vertical components.

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.

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.

*expand content*

*This lesson is great! My students are going to love the connection to real life in our physics class :) | 7 months ago | Reply*

*Responding to Chelsea Gallagher*

Hi Chelsea, I added these documents to the narrative. Hope it helps!

| 3 years ago | Reply

Could you provide an answer key for the Airplane Task and notes worksheet?

| 3 years ago | Reply*expand comments*

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: Triangles That Are Wrong Because They Are Not Right
- LESSON 2: The Law of Sines: More than Meets the Eye
- LESSON 3: Trigonometry from a Geometric Perspective
- LESSON 4: Airplane! - An Introduction to Vectors
- LESSON 5: Making Vector Operations Transparent
- LESSON 6: Formative Assessment: Formulas and Vectors
- LESSON 7: Resultant Vectors
- LESSON 8: Trigonometric Form of Complex Numbers
- LESSON 9: De Moivre's Theorem
- LESSON 10: Unit Review: Additional Trigonometry Topics
- LESSON 11: Unit Review Game: Categories
- LESSON 12: Unit Assessment: Additional Trigonometry Topics