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# Solving Problems with Vectors

Lesson 4 of 11

## Objective: SWBAT use vectors to represent a problem then find the resultant.

*40 minutes*

#### Bell Work

*15 min*

Today I begin by putting up an app called Paper Toss. I let students in the class try the application for a few minutes. I then asked **"How does this application relate to vectors?"** Students struggle at first.** "What are the rays on the screen telling us? How do we use them to make a better shot?"** Engagement is increased with this application. Students interested in programming or in physics may see the connection to vectors first. That is because they have dealt with these ideas in their other courses.

We discuss how the line on the trash can represents the direction and speed of the fan while the line on the paper shows the direction we toss the paper. Drawing a diagram of what happens when we toss the paper helps students see how vectors are used to represent navigation, forces and other real world problems.

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After analyzing the Paper Toss application, students are given a problem. Students work in small groups to determine the component form of the vector. I guide groups with questions such as:

- Have you made a diagram?
- Could you make a triangle?
- How can you use your triangle to determine the x component? y component?

I have students share their answers on the board. I come back to the questions I used during the problem solving time so that students have questions they can ask themselves as they solve problem. Many students realize that we are

I now give students a more complicated problem. Again students work in small groups.

Some student do not see how to use the information to write the component form of the vectors. Understanding the difference between the the velocity vector of the plane and the true velocity vector is also confusing. To help students understand the true velocity vector I ask "What would it mean be be true?" Some students

After five minutes, the first question I ask is, "what did you do to start the problem?" The most common reply is to draw a picture or diagram. Building on this starting point, I ask individual students to share his/her process of modeling the problem with a diagram.

Once a diagram is drawn on the board, we work as a class to analyze it. My initial goal is to introduce students to some vocabulary. Once this is accomplished, we will then work to find the answer to the problem. This strategy helps students develop fluency and precision with the language of and operations with vectors.

Pages 2-4 of problem solving vectors.pdf shows how the class worked to understand the question in the example. After students complete the Bellwork problem, I ask them to work in groups on the Vector Problems worksheet.

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#### Closure

*5 min*

At the end of class, I pose the following questions as an informal assessment of the class:

- How do you add vectors?
- What is the resultant vector?
- What is true velocity? How is it different from velocity?
- How do you calculate the magnitude of a vector?

I conclude by asking the students if they have any questions about vector operations that we should focus on in upcoming lessons.

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Introduction to Vectors
- LESSON 2: Component Form of Vectors
- LESSON 3: Operation with Vectors
- LESSON 4: Solving Problems with Vectors
- LESSON 5: Review of Complex Numbers
- LESSON 6: Complex Numbers and Trigonometry
- LESSON 7: Operations of Complex Numbers in Trigonometric Form
- LESSON 8: DeMoivre's Theorem
- LESSON 9: Roots of Complex Numbers
- LESSON 10: Review
- LESSON 11: Assessment