## The New Material Learned from the Readings

After having read the two articles listed on a reference page, I have learned that although irrational numbers are hard to be written in decimal equivalent, they are easy to visualize using a proper drawing technique. One does not necessarily need to do exact measurements when attempting to draw a line having a length of √2 inches, for example. If the Pythagorean theorem is used, the line can be drawn with no serious calculations applied. For that purpose one simply needs to use a right triangle with both feet being 1 inch long. The hypotenuse of the triangle will be exactly √2 inches in length. It is known, that for an irrational number to be written correctly a long list of figures is required: √17 = 4.123105625617661. However, if a right triangle is drawn and both legs are measured carefully, the hypotenuse will match a required number (Lewis, 2007).

Another peculiar fact I have known about is that when one uses the hypotenuse as afoot for a new right triangle with the length of its second foot corresponding to that of a previous triangle, the wheel of Theodorus starts to occur. The more triangles one draws with the help of this concept, the more a picture reminds a wheel. The author uses this method of visualization when explaining his students the principles of implementation of the Pythagorean theorem to irrational numbers. Each time a new triangle is added to the picture the hypotenuse becomes longer. Though its exact length is hard to measure using a straight scale, one does not need to bother with that kind of a task since the line is already drawn owing to the data of a previous figure.

## The Already Known Facts

Regarding the known facts, calculating the area of a square using right triangles that are present within its structure is the method I have been using since my school years. It, therefore, makes no difference whether the length of a side is a regular or an irregular number. Brown and Owens (2009) highlight, “sides of squares are always square roots” (p. 58). As shown in figure 2 of their article, the feet of the right triangles that are located within a ‘tilted’ square are always regular numbers (Brown & Owens, 2009). By adding the areas of all four triangles (plus a small square in the center), one acquires the area of a tilted square. The picture is a bright example of how one learns to expand his or her spatial thinking and implement visualized concepts for tasks resolving.

## Teaching Techniques that Need to Be Improved

If I were teaching a grade where irrational numbers were formally taught, I would probably combine the techniques used in the two articles to deliver information in the form of a visual game. The wheel of Theodorus is a brilliant example of how one uses the Pythagorean theorem to develop an artistic talent within a pupil. I would, of course, use this concept too but with some enhancements applied. My pupils would not draw the triangles; those would be made of mottled paper coming in the form of applications. It is known that the wheel involves placing some of the triangles underneath the others as long as it continues to ‘spin.’ With paper, mock-ups used this problem is easier to solve.

## References

Brown, R., & Owens, A. (2009). Mathematical explorations: Tilted Squares, Irrational Numbers, and the Pythagorean Theorem. *Mathematics Teaching in the Middle School, 15*(1), 57-62. Web.

Lewis, L. (2007). Irrational Numbers Can “In-Spiral” You. *Mathematics Teaching in the Middle School, 12*(8), 442-446. Web.