# Interactive Visualizations for Effective Math Teaching

Mathematica is, in my opinion, the most effective way for teaching mathematics. With this software, one can clarify and visualize concepts, create assignments, and give instantaneous feedback, all within one display. This has the effect of building students’ self-confidence while allowing them to create their own challenges. As a teacher, you can write your lesson plans or programs and, if you chose, you can use Wolfram Alpha for more clarifications.

In this post, I explore some examples of how I use Mathematica when teaching.

#### Example 1: Illustrating the absolute value in a two coordinate system

The absolute value of a real number x, represented by the symbol | x |, is given by:

This definition can be illustrated by the following plot:

**Horizontal translation of the absolute value of x**

Using Mathematica, we can show the coordinate graphs of | x | and |x + d|, where d is a horizontal displacement:

#### Example II: Solving equations involving absolute values

Here we show hints and solutions for two absolute value problems with checkbox selectors.

Problem 1: Find all real values of x that satisfy: | x | — |x + 3| = -3.

Problem 2: Find all real values of x that satisfy: | x | + |x + 3| = -3.

#### Example III: Illustrating horizontal and vertical shifts

#### Example IV: Illustrating involution of a circle and the meaning of Pi

If you are seeking a more advanced exercise for students who are ready for a challenge, try this one.

A string is wrapped around a circle with radius 1 cm. As the string is unwound, its end traces a path called the involute of the circle, as shown in purple in the graph below.

#### Conclusion

These are just some examples of ways I have used Mathematica to visualize mathematical concepts for students. I have written many other programs to create assignments and explanations. By starting early with programming, middle school students are able to create their own output, check their understanding, and build their self-confidence. I look forward to hearing from readers about how you use this versatile tool in your mathematics instruction.

#### About the blogger:

**Abe Gadalla**

Abe Gadalla is a Minnesota-licensed math teacher for levels 7–12, File Folder # 353655. He holds advanced degrees in engineering with special interest in applied mathematics. Abe has a penchant for understanding the practical applications of theoretical mathematics. A German-educated engineer, Abe found his vocation by taking complex principles of the various math disciplines and breaking them down into simple, yet interesting, concepts and making them appealing to math scholars from the middle to high school levels. Abe gains the respect of his mentees by demonstrating a casual mastery of the subject matter. His strategies in transferring the passion of math to young scholars produces motivated students that pursue math long after he has taught them. Abe teaches a variety of mathematics courses and has presented “Effective Teaching with Mathematica” at several conferences.