9-12 Math: Sum and Differences of Vectors

9-12 Math:  Sum and Differences of Vectors

Instructor Directions

Engineering Design Process

Project Description

Name of ProjectSum and Differences of Vectors
Subject AreaMath
Targeted Standards2003 NC Standard Course of StudyPreCalculus: Objective 1.03 Operate with vectors in two dimensions to model and solve problems.Revised NC 4th Math CurriculaPreCalculus: NC.M4.N2.2
Driving Question / Problem / ActivatorHow do resolve vectors to component form and determine resultants?  How do you determine the magnitude and direction of resultant vectors?  How can robots be used to reinforce students’ understanding of these relationships?
Project SummaryStudents will resolve vectors into x and y-components, combine these values to determine the components of the resultant, and determine the magnitude and direction of the resultant.  
Estimated TimeThis lesson is designed to be presented during a standard 90 minute class.
Materials / Resources/Lesson Plans (including link to slideshow if available)Sphero Robots,  Masking Tape, Tape measure,  Sharpie Markers;  Students will need pencils, paper, and calculators.https://drive.google.com/open?id=1kLbfNdFLZQwAjK4xz1iMp9Csyep88N1Hhttps://drive.google.com/open?id=1GSyK6DQacTas-sJ9XDg5OIywMZCujT0S
Tags 

Project Outline 

AskHow do you resolve vectors to component form and determine resultants?  How do you determine the magnitude and direction of resultant vectors?  Can robots be used to reinforce students’ understanding of these relationships?
ImagineHow could this activity be be accomplished using different types of robots?  Other types of robots are more precise in their movements, but Spheros are easy to program and relatively inexpensive.
PlanStudents will calculate travel times and directions necessary to traverse vectors.
CreateStudents will program their robots to traverse the individual vectors, as well as the resultant vectors.
ImproveDetermine improvements to the layout of the work area, sequence of instructions, wording of narrative and instructions
Closure / Student Reflections1) How closely did the robot’s stopping point at the  endpoint of  each resultant vector agree with the sum/difference of the given vectors?3) After observing robots’ response to programmed directions, what program revisions could be made to improve accuracy?2) What sources of error might affect the accuracy of a robot’s travel?
Possible Modifications / ExtensionsEvaluate the effect of travel speed on the accuracy of travel.  Evaluate the effect of different surfaces on the accuracy of travel.

Evaluation (Pre/Post)  

At the start of the activity:

 

After the activity:

1) How far apart were the endpoints of the resultant vectors from the final endpoint of the component vectors?          

Activity A                Activity B                Activity C

 

2) Evaluate the following sources of error on the accuracy of a robot’s travels.

a.  How does a robot’s initial alignment affect the accuracy of each activity?

 

 b.  Speed; Suppose a robot’s actual speed is faster or slower than accounted for in your calculations.  Determine the error in distance traveled over 15 seconds for the following speed errors. (Show two sample calculations.)

 

Speed ErrorTravel ErrorSpeed ErrorTravel Error
+ 0.1 in/sec –  0.1 in/sec 
+ 0.15 in/sec –  0.15 in/sec 
+0.5 in/sec – 0.5 in/sec 
+ 1 in/sec –  1 in/sec 

 

c.  Direction/Angle; Suppose a robot’s direction differs from the “correct” direction.  Given a triangle with two sides that are each 15 ft, determine the error in the “third” side of the triangle that results from the following errors in the “included” angle. (Show two sample calculations.)

 

Direction Error“3rd Side” ErrorDirection Error“3rd Side” Error
+ 0.5O –  0.5O 
+ 1O – 1O 
+ 3O –  3O 
+ 5O – 5O 

 

 3) What other sources of error might affect the results of these activities?

 4) Using the speed settings from the three triangles, determine the roll durations and headings/bearings necessary to verify the resultant sum of the following vectors.

     Given:                

          Find the resultant:

 

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