February 2012 - Page 5 of 7 - Math and Multimedia

Introduction to the Complex Plane

February 12, 2012 GB High School Mathematics

Imaginary numbers had no use when it was invented except for intellectual exploration until it was linked to coordinate geometry. The complex number $a + bi$ can be plotted as the ordered pair $(a,b)$ where $a$ is the real part and $b$ is the imaginary part. Therefore, the complex numbers $3 + 2i$ , $3- 2i$ , $-2 + 0i$ , and $0 - i$ can be plotted as $(3,2)$ , $(3,-2)$ , $(-2,0)$ , and $(0,-1)$ in a “coordinate plane” respectively as shown below. In the figure, we can see that the origin is $0 + 0i$ and that $3 - 2i$ , the complex conjugate of $3 + 2i$ , is its reflection along the x-axis (Can this be generalized?).

The plane where the complex numbers are plotted above is called the complex plane.

In the complex plane, we can observe that all numbers of the form $(a,0)$ are real numbers and these numbers are represented by the horizontal axis. The numbers of the form $(0,bi)$ are all imaginary numbers and it is represented by the vertical axis. Thus, we can call them the real axis and the imaginary axis respectively.

In addition, we can conclude that all real numbers are complex numbers since for all real numbers $a$ , $a = a + 0i$ . So, the set of real numbers is a subset of the set of complex numbers.

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Popularizing Lesson Study

February 10, 2012 GB Pedagogy and Teaching

The reason that I was away for two days was that I observed the lesson implementation in Nueva Ecija High School (NEHS). It was part of lesson study, one of the components of the 2-year project of our institute and NEHS.

Lesson study is a professional development program for teachers that originated in Japan. In lesson study teachers collaborate with one another in developing and implementing a lesson. If you have not heard about this type of professional development program, it is now gaining popularity worldwide and is already practiced by elementary school and high school teachers in many countries.

The process in lesson comprises of the following steps:

Defining a teaching problem based upon student needs
Lesson Study planning, with the student and the teacher as the focus
Focusing the lesson on student thinking, learning, and misconception
Evaluating the lesson’s impact on student learning and reflecting on its effect
Revising the lesson based upon the data collected
Teaching the revised lesson to a new class of students
Evaluating and reflecting
Sharing the results.

The lesson developed in lesson study is called ‘research lesson.’ » Read more

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Carnival 20 deadline of submission

February 9, 2012 GB Blog Carnival

The Mathematics and Multimedia Blog Carnival is on its 20th edition and will be posted at GeoGebra Applet Central on February 20, 2012. If you have a math blog and have written an article about school mathematics, mathematics teaching, or integration of technology in teaching math, then you may submit your post on or before February 18. Submitting your post to a math carnival is one of the best ways to expose it to more readers.

For those who are new to math carnivals and those who are interested in hosting, you may want to read the following posts:

Here are also the latest editions of the carnivals for your reading pleasure.

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