# Curricular Ideas

As noted in the table of contents, the topics below are placed in a possible order of delivery that makes the most sense to me. However, all of this is subject to change with classroom implementation, hopefully starting in 2022. In other words, this is less of a long-term or unit plan and more of a loose outline of topics I hope to cover, including potential lines of student research. You’ll also notice that much of this is still a work in progress. With that said, each topic in bold is followed by bullet points representing associated research tasks. Remember, these are not meant to simply be printed, then handed to students, but should be used as a guide that you, and your students, are free to deviate from if a more interesting question emerges while in pursuit of an answer.

The Need for a Coordinate System: Ultimately, a coordinate system is about establishing a point of reference by which the position of all other objects are established. The focus here is on the x, y, and z-axes.

• How Many Paths?: Research a problem that will inspire the need for factorials.

The Infamous Pythagorean Theorem: Most people can recite the PT by memory, but explore a proof for why it works.

• Midpoints: With the PT, there is an opportunity to develop the strategies to find the midpoints of line segments and verify results.
• From Pythagoras to Fermat: Fermat’s Last Theorem was only recently solved and is a logical extension from the PT.
• Traveling Salesman: Another open-ended problem, this can inspire some unique strategies with a wide range of applications.

Area and Perimeter: Area and perimeter are often the easiest to design applications for, therefore this section focuses on understanding what these terms actually represent and the derivation of various formulas.

• Irregular Features: Not all shapes are circles or polygons, therefore students will be challenged to find methods for approximating area. Google Earth has an interesting tool for students to play with.
• Gauss Circles: Moving beyond applications, here is a pure mathematics investigation into another open problem.

Point(s) of Intersection: Thinking about the paths people take wandering the landscape, an obvious question is when might people meet each other and/or cross paths. Explore what it means to find a point of intersection and strategies for doing so.

• When Strange Answers Arise: Investigate the mystery of seemingly nonsensical answers when solving for points of intersection.
• When Formulas Fail: Although many can recite the quadratic formula form memory, what about the cubic formula? The quartic formula? Can an algebraic formula find the roots for polynomials of any degree?
• Centers of a Triangle: Explore properties regarding points of intersection within a triangle and look into potential applications.

Graph Theory: Learn some of the basic notation, terminology, and applications.

• Seven Bridges: Explore the problem solved by Euler and move into a bit of graph theory.
• Graph Coloring: Connect students to one of the open problems in mathematics.

Degrees Versus Radians: Degrees and radians are compared in a way that explains when one may be more useful than the other.

The Rules of Traditional Trigonometry: This dives into the geometric definitions of sine and cosine, then derives the law of sines and law of cosines.

• Triangulation: Trigonometry is a powerful tool for establishing the position of an object when two other points are known. Students are challenged to prove a couple of formulas, then explain the elements of a proof.
• Finding Latitude: Develop a technique for finding one’s location while on the open ocean.

Trilateration-How GPS Works: Triangulation is a useful tool, but modern GPS technology sends out transmissions, records distances, then locates position utilizing circles. Check this page out for a mathematical explanation.

Surface Area and Volume: Also easier to design applications for, therefore this section focuses on understanding what these terms actually represent and the derivation of various formulas.

• Sphere Packing: How many spheres can be packed into a space of given dimensions (to be added).

Polar and Spherical Coordinates: The Cartesian Plane is great, but sometimes it is much easier to think in terms of angles and radii.

• Conversions: Research methods for converting between cartesian and polar coordinates, then use Desmos and Geogebra to verify results. Then move onto conversions between coordinates in (x, y, z) format to spherical coordinates.
• Cylindrical Coordinates: What might a cylindrical coordinate system look like?

Spherical Trigonometry: What if, rather than a traditional plane, the plane is the surface of a sphere? The law of sines and the law of cosines must be adjusted to conform to this new reality.

• Changing Relationships: Shockingly, a spherical triangle has angles that can add up to more than 180 degrees. Have students investigate some of the changing relationships.

Producing Figures in 3-Dimensional Space: Building upon the equations for ellipses, parabolas, and hyperbolas, this page will explain how to derive the formulas for their 3-D counterparts.

• So You Said Conic Sections Come From Slicing a Cone…?: Students will prove that the intersection between a flat plan and a cone actually produces each of the conic sections. This includes a video from one of my favorite popularizers of mathematics, Grant Sanderson.
• Building the Solar System: Now that students understand polar coordinates and conics sections, they should be able to build a simple, animated model of the solar system in Desmos or Geogebra.

Communicating Topography: Topographical maps have to communicate three dimensions on a two-dimensional surface. Read this page to find out how this is done.

• Build the Landscape: Given a topographical map, students will be asked to reconstruct the landscape in Geogebra.

Parametric Equations: Rather than describing curves in terms of x, y, and z, we can actually use trigonometric functions, radii, and angles. Here, we will investigate how this works, including a discussion of parameters.

• Creating Planet Earth: Research the building of a spherical planet with latitude and longitude using Geogebra. However, Earth technically isn’t a sphere…
• Traveling Around Other 3D Figures: While great circles show the fastest way from one point to another around a sphere, shapes like cubes are far less obvious.

The Power of Tangency: Although the Earth is a sphere, 2D maps of a region are a good approximation thanks to the powerful mathematical concept of tangency.

Map Projections: Portraying a spherical planet on a two-dimensional world map is an interesting mathematical pursuit. Find an exploration of the popular Mercator projection and a method for communicating distortions.

• Other Projection Methods: Students will investigate other types of projections, including their strengths and weaknesses.
• Measuring Distortion: No matter which technique is used to project the earth onto a 2-D map, features will be distorted. Students will investigate a method for measuring and visualizing that distortion.

Matrix Algebra: Matrices are really just lists of numbers organized into columns and rows, yet they are powerful tools for solving equations, handling multi-dimensional problems, and rotating planes, to name just a few of their uses. Learn more about operations with matrices and some of their basic properties.

• Rotating Planes with Matrices and Trig: In the “Oh No! What About Elevation Task!?!,” the final problem alluded to the use of matrices. Use what you learned on the Matrix Algebra page to expand your research.