
Course 4 Unit 2  Vectors and Motion
©2010
In Course 4:
Preparation for Calculus, geometry and algebra become increasingly
intertwined. Students develop understanding of twodimensional vectors
and their application and the use of parametric equations in modeling
linear, circular, and other nonlinear motion. In addition, students
intending to pursue programs in the mathematical, physical, and biological
sciences, or engineering extend their ability to visualize and represent
threedimensional surfaces using contours, cross sections, and reliefs;
and to visualize and sketch surfaces and conic sections defined by
algebraic equations. They also extend their understanding of, and
ability to reason with, trigonometric functions to prove or disprove
trigonometric identities and to solve trigonometric equations. They
also geometrically represent complex numbers and apply complex number
operations to find powers and roots of complex numbers expressed
in trigonometric form. (See the CPMP
Courses 14 descriptions.)
Unit Overview
Motion is a frequently
occurring aspect of our lives. It is natural to ask how motion can be
modeled mathematically. Two mathematical tools are introduced in this
unit that help students model motion. The first is the vector, which
is introduced initially as a free vector (directed line segment) that
is drawn wherever it may be needed and later as a position vector attached
to a specific point. It is the position vector representation that leads
to the introduction of the second tool, which is parametric representation
of locations specified by vectors. The set of these locations forms the
graph of the motion as a function of a third variable, usually time.
The parametric representations and a graphing calculator or computer
graphing software allow students to see linear, projectile, circular,
and elliptical motions of objects and to analyze the resulting paths.
Lesson 1 develops vectors and their
geometric representation, scaling of vectors (including the opposite
of a vector), and combining vectors by addition. These concepts are developed
in the context of navigation and applied to other situations. In Lesson 2,
vectors are analyzed in a coordinate system. The coordinate representation
of vectors is then used to prove geometric linear motion algebraically
using parametric equations. Lesson 3 extends simulation of motion
using parametric equations to model projectile motion and circular and
elliptical motion.
Objectives
of the Unit
 Describe
and use the concept of vector in mathematical, scientific,
and everyday situations
 Represent
vectors geometrically and operate on geometric vectors
 Describe,
represent, and use vector components and operations synthetically
and analytically
 Investigate
and justify general properties of vectors and vector operations
 Provide
vector proofs of properties of triangles and parallelograms
 Use vector
concepts to parametrically represent linear, projectile,
circular, and elliptical motions in a plane
 Analyze
motions using parametric models

Sample
Overview
The sample investigation
below is Investigation 2 from Lesson 2. In the first investigation
of this lesson, students explored coordinate representations of vectors.
In this investigation, after exploring operations on vectors, students
examine how vectors can be used to establish geometric relationships.
Instructional
Design
Throughout the curriculum,
interesting problem contexts serve as the foundation for instruction.
As lessons unfold around these problem situations, classroom instruction
tends to follow a fourphase cycle of classroom activities—Launch,
Explore, Share and Summarize, and Apply. This instructional model is
elaborated under Instructional Design.
View the
Unit Table of Contents and Sample Lesson Material
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