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NetLogo Models Library:
Sample Models/Mathematics

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Color Fractions

[screen shot]

If you download the NetLogo application, this model is included. (You can also run this model in your browser, but we don't recommend it; details here.)

WHAT IS IT?

Color Fractions displays the decimal form of a fraction as a pattern of colors. This lets you investigate how the resulting patterns relate to the original fraction.

HOW IT WORKS

The program computes the digits of the decimal the same way you would do it using long division on paper.

It takes the numerator and divides it by the denominator; the result is a digit. The remainder of the division, we multiply by 10 and take as the new numerator, then repeat.

For example these would be the steps for computing 1/7:

  • 7 goes into 1 0 times; 10 times remainder is 10
  • 7 goes into 10 1 time ; 10 times remainder is 30
  • 7 goes into 30 4 times; 10 times remainder is 20
  • 7 goes into 20 2 times; 10 times remainder is 60
  • 7 goes into 60 8 times; 10 times remainder is 40

And so on forever. Thus the result is 0.1428...

To make the pattern of colors, each digit, 0 through 9, is shown in its own color. For example 0 is gray, 1 is light red, 2 is orange, etc.

HOW TO USE IT

Press the SETUP button, then press GO.

Now adjust the numerator and denominator sliders to make any fraction.

You can also use the WIDTH slider to change the total number of digits per row. Changing the width can make patterns easier or harder to see.

THINGS TO NOTICE

The same fraction always gives the same pattern.

Some patterns stop after a few digits, while others repeat forever. (Do all fractions repeat?)

Do two fractions ever give the same pattern?

THINGS TO TRY

What kinds of patterns can you generate with fractions? Can you create a checkerboard? Can you make vertical stripes?

What happens if you try different numerators with the same denominator? What do the patterns you get have in common? Try this experiment with several different denominators.

The "period" of a fraction is how many digits repeat to make the pattern. For example, 1/3 is 0.33333... which repeats the same single digit over and over again, so the period is 1. Look at the pattern for the fraction 1/7; the period is 6. And so on.

How is the period of a fraction 1/n related to n? How are the periods of the fractions family 1/7, 2/7, ... related?

How many fractions can you find that have the same number of repeating digits?

What fraction best approximates pi (3.141592...)?

EXTENDING THE MODEL

Change the colors so odd numbers are all one color and even numbers are another color.

Change the colors so you only use one color, but vary the lightness/darkness.

Modify the program so you can visualize square roots and pi.

Extend the program so you can have a full fraction Calculator such as the one in the references.

NETLOGO FEATURES

Patch labels are used to show the digits.

You can use the sort primitive to created a list of patches sorted left-to-right, top-to-bottom. That is necessary in this model because we need the patches to execute in the same order at every tick, rather than a different random order every tick as would happen if we just said ask patches.

RELATED MODELS

Division

CREDITS AND REFERENCES

This model is adapted from: Color Calculator Information for teachers and Color Calculator.

This model implements activities originally published in: Nathalie Sinclair, Rina Zazkis and Peter Liljedahl. Number Worlds: Visual and Experimental Access to Elementary Number Theory Concepts. International Journal of Computers for Mathematical Learning, Volume 8, Number 3, pages 235 - 263, January 2003.

HOW TO CITE

If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

Please cite the NetLogo software as:

COPYRIGHT AND LICENSE

Copyright 2005 Uri Wilensky.

CC BY-NC-SA 3.0

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

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