Mathematics

Visual Organizers in Mathematics

We use math in many ways to organize and present data.

1. First we use numbers in several ways. We create lists like this one. We also use numbers in scales as rating something on a scale of one to ten. On many devices like the common thermometer, we use numbers to define the many results. We can also arrange objects by size.

2. We also use numbers on a logarithmic scale. In these scales, each number is ten times the previous number.

The most familiar of these is the Richter Scale for measuring earthquakes.
The largest earthquake ever measured was a 9.5 quake in Chile in 1960. Assume the Richter scale goes up to 10. If you used counting numbers instead of logarithms, we would need to rate them from one to one billion. Most of us have an easier time comprehending the numbers one to ten. We don’t need to understand what a logarithm is. We just know that an earthquake of 7.0 is ten times as powerful as an earthquake of 6.0.

Another example of a logarithmic scale would be the pH scale used to define the acidity or alkalinity of a substance.

A simple pH scale shows acidity or alkalinity of common substancesIf we used counting numbers here, the scale might show only the ranges of the more powerful acids and bases with very little discrimination between distilled water and acid rain. The other possibility would be rating them from one to one hundred trillion just to go from the scales rating from one to ten. Because we now know substances that are far more acidic or alkaline, this ranges would need to be expanded much further.

Like with earthquakes, we simply need to remember that a substance with a pH of 5 is ten times more acidic than one with than one with a pH of 6, one hundred times more acidic than one with a pH of 7(considered neutral.).  A substance of pH 12 is one hundred times more alkaline than one of pH 10. As you look at the chart, you might say stomach acid is right next to lemon juice. I thought it was a lot stronger than that. Then remember this means stomach acid is ten times as acidic as the lemon juice.

3. We can also point to ways other areas of math are commonly used to organize data. Statistics, with it’s variety of charts and graphs is so important, it forms a separate category here.  Geometric shapes are used to organize data in many ways. One interesting use is with Venn Diagrams to show overlapping information. They can be used for compare and contrast charts.

4. One area of math has only recently been used to organize data. This is the group of graphs called Fractals.
For many years mathematicians created graphs of equations on x and y coordinates (graph paper). Since the 17th century, mathematicians explored the idea of fractals. Many students are familiar with the simple forms including the Koch Snowflake and the Sierpinski Triangle, showing how, with many iterations (doing the same thing over and over, you create an image that is similar when considered at different levels.

In i975, Benoit Mandlebrot  introduced the term fractal. We now have astonishing, brightly-colored computer-created patterns derived from  equations with fractional dimensions.. Best known are the Mandlebrot set and the Julia Set. These studies have proved valuable in many fields of knowledge.

Mandelbrot Fractal by Markus GannJust as amazing as their beauty and magical repetitions of patterns at different levels, are the many practical applications that have been found.

Seismologists use them to study earthquakes. In medicine, they have been used to study heartbeats, to detect brain tumors, and to study wear on teeth. They are also used in the study of turbulence and in enlarging digital photos.

For a YouTube experience with fractals try one of these

Julia Set Fractal Part 1 of 3 by florianchurch

Fractal Zoom Mandelbrot Corner by Gooozz

Fractal Zoom by Crazy Mix

For those seriously interested in fractals, try a 17 minute lecture: Ron Eglash Discusses Fractals in African Village Architecture.

How Might Students Use Mathematics to Organize Data?

Note: You will find it easiest to use math to organize date in the sciences where you have measurable data from an experiment or survey.

  • You might number subjects or concepts in order of importance.
  • You might arrange events in the order they occurred, using a Timeline (more on this on another page.)
  • You might use Venn Diagrams to show relationships.
  • You can measure your data.
  • You can study the rate at which events or change takes place.
  • You can explore the probabilities of a certain event.
  • You can discover patterns and relationships in your data that can be shown using geometric shapes
  • You might discover patterns  and relationships in your data that can be summed up in an equation.

When I was teaching, I liked to show the video called Powers of Ten. It displayed how huge the differences were between ten to the third power, for example and ten to the fourth or fifth power. It also helped us recognize what immense differences there were between the size of a quark and the size of an atom, or between the size of the earth and the size of the solar system. To see this go to  http://scaleofuniverse.com and click on the tab for Powers of Ten. My suggestion is that you first picture one square meter. That is 10° meters. Now picture a space that is 10 meters squared…. 10 meter on each size. This is 10¹ meters. Then estmate the powers of ten for the earth, for our sun, and for the known universe. Estimate the powers of ten for the size of an ant, for an e coli bacteria, for a proton.  Then watch the video and see how well you did.  As you watch, notice the powers of ten shown on the right side of the screen.

NOTE: The website is at scaleofuniverse.com  NOT scaleof THEuniverse.com

At that same website, the  Scale of the Universe Cary and Michael Huang. Again, as you marvel at the immensely large and small parts of our universe, be aware of the powers of ten.

Next: Statistics

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