Math Education Concepts

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Radians and Degrees…

I really enjoy reading posts by Jason Marshall, PhD, The Math Dude.  He has a very simple way of explaining complex concepts.  The post that caught my attention today is What are Radians and Degrees?.  If you teach or study geometry or trigonometry you will encounter radians and degrees.  The Math Dude explains the difference between them and how to convert from one to the other.

Click here to read more about this fascinating concept in mathematics…


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Teaching Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse trigonometric functions (in brief).  Trigonometric functions are functions of an angle measure.  They are primarily used to find lengths of the legs of a triangle (or some triangular relationship).  Inverse trigonometric functions are functions of the ratio of lengths of a triangle.  They are primarily used to find the corresponding angle of the ratio of the legs of a triangle.  There are other uses, but I will keep it brief for this blog.

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Every semester, I have to put more effort into explaining inverse trigonometric functions.  Although the concept is the same, the students, and their perceptions, change.  There are only so many ways I can think of to explain this concept.  So I decided to get some help from several online resources.  Of course, I suggest that my students do the same, but that’s another blog, for another day.

Here are a few links I came across that should be helpful to students and teachers alike, looking for alternative ways to explain inverse trigonometric functions.

WolframMathWorld

Khan Academy

Randy Anderson (YouTube)

Paul’s Online Math Notes (Paul Dawkins)

How do you teach inverse trigonometric functions to your students?

 

 


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Applications of Trigonometric Functions

Trigonometric functions are functions of angles and are useful for finding the lengths of the sides and measures of the angles of triangles (primarily right triangles).  They are also useful for describing harmonic or periodic motion, such as sound waves.

Most students either love or despise trigonometric functions.  I happen to enjoy them.  They are mysterious, yet simplistic functions.  When you get them, you get them!!!

When teaching the trigonometric functions, I approach them in one of two ways: using right triangle trigonometry (the most common) or using components of the slope of a line (rise, run, slope), etc.  I recently started using the second approach to see if students new to trigonometry would understand the functions easily.  I’m still working that out.

 

 

 

 

 

 

 

Trigonometric functions are applied in astronomy, geography, engineering, physics, and architecture.  Here are a few examples:

  1. Distance between planets
  2. Height of mountains
  3. Dimensions of land

The history of trigonometry tells me that it all started with the “stars” or spherical geometry.  Linear algebra followed and we now have a very comprehensive system that revolves around the trigonometric functions.  Every semester, I learn something new about these functions and make a note to myself to learn more!

NOTE: This blog does no justice for the true value of trigonometric functions.  My goal here is to inspire you to research trigonometry for yourself and find out why they are so intriguing.

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Preparing for Calculus

Calculus is a specified field of mathematics.  Limits, derivatives, integrals, and infinite series are all applied to various types of functions (i.e., linear, quadratic, exponential, trigonometric, etc.).  They each depend on the understanding of basic arithmetic as well as specific concepts of other branches of secondary level mathematics such as algebra, geometry, and trigonometry.  Prior to taking a calculus course students should develop specific skills in each of these areas of study.

Below, I briefly discuss key skills and concepts required from three branches of mathematics (algebra, geometry, and trigonometry) to prepare students for calculus.  I discuss this material as it relates to differentiation, a fundamental component of calculus.

Algebra (Functions)

Differentiation converts formulas into other formulas.  These formulas are functions.  Algebra is the study of functions.  Since functions are the object of differentiation, it is imperative that students entering into a course of study in calculus have a thorough understanding of functions.

Many secondary students can follow instructions to substitute a value into an equation, but the study of functions will provide a greater understanding of why the method of substitution works and why a relationship exists between the parameters.

Geometry (Graphs)

The result of differentiation is a derivative.  Derivatives are used to calculate the slope of a line tangent to a point on the graph of a function.  A student can find an estimated value using the slope of a line secant to the same graph, but calculus makes it possible to find the exact value.  Students must have a thorough understanding of graphs, slopes, and functions to understand the use of derivatives with respect to graphs of functions.

Trigonometry (Angles, Periods of revolution)

Trigonometry is the study of trigonometric functions.  Trigonometric functions are used to calculate measures involving triangles, periods of revolution, etc.  During the course of a calculus class students will encounter problems involving trigonometric functions which are useful for explaining periodicity.  Students studying calculus should understand trigonometric functions, their properties, and their graphs.

As discussed above, students preparing to take calculus should have a specific set of skills relating to calculus content.  The major subjects required for the development of these skills, and the understanding of the related concepts, are algebra, geometry, and trigonometry.  More specifically, students need to have a thorough understanding of functions, graphs, slopes, and trigonometric functions, to name a few, in order to effectively prepare to enter into a calculus course.

Although students have the benefit of the use of technology to solve many problems, they must have an understanding of the relationship between the functions used, their graphs, and the derivative so they can understand the output generated from the use of technology.  Simply plugging values into a technological tool does not provide this level of understanding.  In most cases, students will not have access to technological tools in college level math courses (see “Calculators Not Allowed” on this blog site), so it’s best to learn and understand the concepts without these aids.

Do you have any other ideas about concepts students should learn prior to taking a calculus course?