Pascal's Prime Number Multiple Fractals: 2 (and odds),3,5,7,11,13,17

My favorite discovery in Pascal’s are the fractals first found by Sierpinski in 1915 (almost 300 years after Pascal); it is referred to as Sierpinski’s Triangle and is created by coloring in the even numbers (numbers that end in 0,2,4,6,8). It is also quite beautiful to color in the odd numbers (numbers that end in 1,3,5,7,9); you will get a negative image of Sierpinski’s Triangle. 

Then I thought about what happens if you color in other prime number multiples; would it form a fractal (a self-similar object) or just a pretty symmetrical picture? It turns out that if you color in prime number multiples such as 3, 5, 7, 11, 13 and 17, you always get a beautiful fractal. Is this true for all prime numbers?

The objective of this lesson was not only to introduce the concept of fractals but to teach them the divisibility rules for these critical primes. This exercise looks like an art project but the children are reinforcing their knowledge of divisibility rules dozens of times for each fractal. The divisibility rules are as follows:

A number is divisible by 2 if its last digit is also (i.e. 0,2,4,6 or 8).

A number is divisible by 3 if the sum of its digits is also. Example: 534: 5+3+4=12 and 1+2=3 so 534 is divisible by 3.

A number is divisible by 5 if the last digit is 5 or 0.

K-2nd grade should be comfortable doing the multiples of 2 fractal as well as the odd number negative fractal on pages one and two of the pdf. Multiples of 3 will be wonderfully challenging when they get to 3 and 4 digit numbers. Remember, all they are doing is adding the digits of the number to see if the sum is a multiple of 3. 

3rd-6th graders should also do the multiples of 2 and 3 but should move on to multiples of 7, 11, 13, and 17.

Test for divisibility by 7. Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary. Example: 826. Twice 6 is 12. So take 12 from the truncated 82. Now 82-12=70. This is divisible by 7, so 826 is divisible by 7 also.

There are similar rules for the remaining primes under 50, i.e. 11,13, 17,19,23,29,31,37,41,43 and 47.

Test for divisibility by 11. Starting with the first digit of the number, take every other digit of the number (the first digit, the third digit, the fifth digit, etc) and add them up. Now take the digits you didn’t use the first time (the second digit, the fourth digit, etc.) and add THEM up. Now subtract these two sums from each other. If the difference between the two sums is divisible by 11, then the number is divisible by 11.

A lot of the time, two sums will be equal, so their difference equals 0, which is divisible by everything, so certainly is divisible by 11. (note: this rule only works for divisibility by 11!)

Test for divisibility by 13. Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary. 

Example: 50661-->5066+4=5070-->507+0=507-->50+28=78 and 78 is 6*13, so 50661 is divisible by 13.

Test for divisibility by 17. Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. Apply this rule over and over again as necessary. 

Example: 3978-->397-5*8=357-->35-5*7=0. So 3978 is divisible by 17.

The pdf allows students to try all 8 fractals as well as one on their own. 

Cautionary Note: often the children will color in the top row of a triangle without testing the numbers for divisibility (one example is 48620 which is not a multiple of 3 but the children often assume that it must be since it forms a larger triangle below the first few iterations of the multiples of three fractal; 4+8+6+2=20, not a multiple of 3).