published by
the Center for Innovation in Engineering and Science Education

This is a high school instructional unit that features nine lessons relating to vectors. Users build understanding of vector properties as they learn airplane navigation. Problem-based learning activities include reading real-time weather maps, tracking airplanes flying in U.S. skies, calculating vector components, analyzing effects of wind velocity, and completing training similar to a private pilot certification program. The unit culminates with a pilot flight test. Participants also have access to help from experts at the Polaris Career Center. Comprehensive teacher guides, student guides , reference materials, and assessments are included.

This resource was developed by the Center for Innovation in Science and Engineering Education (CIESE). Participation is cost-free with teacher registration.

9-12: 2B/H3. Mathematics provides a precise language to describe objects and events and the relationships among them. In addition, mathematics provides tools for solving problems, analyzing data, and making logical arguments.

2C. Mathematical Inquiry

6-8: 2C/M2b. Using mathematics to solve a problem requires choosing what mathematics to use; probably making some simplifying assumptions, estimates, or approximations; doing computations; and then checking to see whether the answer makes sense.

9-12: 2C/H3. To be able to use and interpret mathematics well, it is necessary to be concerned with more than the mathematical validity of abstract operations and to take into account how well they correspond to the properties of the things represented.

4. The Physical Setting

4F. Motion

6-8: 4F/M3a. An unbalanced force acting on an object changes its speed or direction of motion, or both.

9-12: 4F/H2. All motion is relative to whatever frame of reference is chosen, for there is no motionless frame from which to judge all motion.

9-12: 4F/H8. Any object maintains a constant speed and direction of motion unless an unbalanced outside force acts on it.

9. The Mathematical World

9B. Symbolic Relationships

9-12: 9B/H4. Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.

12. Habits of Mind

12B. Computation and Estimation

9-12: 12B/H2. Find answers to real-world problems by substituting numerical values in simple algebraic formulas and check the answer by reviewing the steps of the calculation and by judging whether the answer is reasonable.

12C. Manipulation and Observation

6-8: 12C/M2. Use computer databases to store and retrieve information.

Common Core State Standards for Mathematics Alignments

Standards for Mathematical Practice (K-12)

MP.1 Make sense of problems and persevere in solving them.

MP.6 Attend to precision.

Expressions and Equations (6-8)

Represent and analyze quantitative relationships between
dependent and independent variables. (6)

6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

Solve real-life and mathematical problems using numerical and
algebraic expressions and equations. (7)

7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

High School — Number and Quantity (9-12)

Vector and Matrix Quantities (9-12)

N-VM.1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

N-VM.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

N-VM.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

N-VM.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

High School — Geometry (9-12)

Similarity, Right Triangles, and Trigonometry (9-12)

G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.^{?}

Common Core State Reading Standards for Literacy in Science and Technical Subjects 6—12

Range of Reading and Level of Text Complexity (6-12)

RST.9-10.10 By the end of grade 10, read and comprehend science/technical texts in the grades 9—10 text complexity band independently and proficiently.

<a href="https://www.compadre.org/portal/items/detail.cfm?ID=8382">Center for Innovation in Engineering and Science Education. Navigational Vectors. Hoboken: Center for Innovation in Engineering and Science Education, January 31, 2008.</a>

Navigational Vectors. (2008, January 31). Retrieved November 2, 2024, from Center for Innovation in Engineering and Science Education: http://ciese.org/curriculum/vectorproj/

Center for Innovation in Engineering and Science Education. Navigational Vectors. Hoboken: Center for Innovation in Engineering and Science Education, January 31, 2008. http://ciese.org/curriculum/vectorproj/ (accessed 2 November 2024).

Navigational Vectors. Hoboken: Center for Innovation in Engineering and Science Education, 2005. 31 Jan. 2008. 2 Nov. 2024 <http://ciese.org/curriculum/vectorproj/>.

@misc{
Title = {Navigational Vectors},
Publisher = {Center for Innovation in Engineering and Science Education},
Volume = {2024},
Number = {2 November 2024},
Month = {January 31, 2008},
Year = {2005}
}

%T Navigational Vectors %D January 31, 2008 %I Center for Innovation in Engineering and Science Education %C Hoboken %U http://ciese.org/curriculum/vectorproj/ %O text/html

%0 Electronic Source %D January 31, 2008 %T Navigational Vectors %I Center for Innovation in Engineering and Science Education %V 2024 %N 2 November 2024 %8 January 31, 2008 %9 text/html %U http://ciese.org/curriculum/vectorproj/

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