Shodor’s Interactive includes interactive applets for student discovery.
There are MANY applets. I randomly chose one that had a model of Pascal’s Triangle. The user could either set their own divisor or have the computer choose a random one. Once the divisor was set, the user shaded each number in the triangle that, when divided by the chosen number, would have a remainder of zero. The user quickly saw a pattern emerge.
This activity would be a good one to practice rules of divisibility.
A number is divisible by:
2, if it is even
3, if the sum of the digits is divisible by 3
4, if the last two digits are divisible by 4
5, if it ends in 0 or 5
6, if it is even and divisible by 3 (see above)
7, no good rule
8, if the last three digits are divisible by 8
9, if the sum of the digits is divisible by 9
10, if it ends in zero
11, if the sum of every other digit equals the sum of the OTHER ever other digits (for example, 12342 is divisible by 11 because 1 + 3 + 2 = 6 and 2 + 4 = 6
I tried it out with divisibility by 11 and discovered a new rule for divisibility by 11! The rule stated above does work for many numbers, but ALL numbers that are divisible by 11 don’t follow this rule. The above image shows my result after I had chosen all number that followed the rule stated above.
The applet said that there were still 5 numbers that were divisible by 11. Since I had used every row that was available (by choosing “increase depth” until it wouldn’t increase any more), I knew that the five unshaded numbers must also be divisible by 11. I looked at the numbers still in blue. They were 924, 715 (twice), and 1716 (twice). These three numbers did not follow the rule that I was using: 9 + 4 did not equal 2, 7 + 5 did not equal 1, and 7 + 6 did not equal 1 + 1. What I noticed, however, was that if I added 9 + 4 and SUBTRACTED 2, I got 11. I tried this with the other three digit number: (7 + 5) – 1 = 11! I was onto something. What about the four digit number? If I added 7 + 6 and subtracted 1 + 1, I also got 11.
So were there two rules? I put the number 15928 into the calculator. I chose this number because (1 + 9 + 8) – (5 + 2) = 11. When I divided it by 11, I got 1,448, which meant that this rule did seem to work. I tried one more very large number, 658,152, since (6 + 8 + 5) – (5 + 1 + 2) = 11. Sure enough, I got 59,832.
Now I’m sure that this rule is out there already in many places, and many of you who read this post will say, “Of course, that rule works…duh!” However, it was new to me! I felt like a student again discovering something new to ME. I knew there must be a pattern. I saw a pattern in the three numbers. Then I tried some other numbers and it worked. The next step…if I choose to take it on…would be to prove algebraically WHY these rules work. I will save that proof until another time.
In the meantime, I am excited to have found a creative use for a particular activity…and, while trying it out, I was able to expand my knowledge. I can see this used with students in just this way. Give them the first rule stated. Have them go out to the maximum number of rows….and see that there are five that ARE divisible by 11 that don’t follow the rule. Then see if they can discover their own rule and come up with further examples.
Now, I want to go back and try to figure out the rule for 7, which I’ve always been told was too confusing to memorize.
I love math!
Okay, so I couldn’t help myself and went to see if I could see a pattern for 7. I didn’t see anything, but I did play further with the activity. After the user finds all of the multiples of 7 (those numbers where, when divided by 7, the remainder is zero), the next task is to find all of the numbers that, when divided by 7, the remainder is 1. The numbers are shaded in a different color…and not surprisingly show a pattern. The next task is finding those with remainder 2, 3, 4, 5, and 6. Below is the image of the completed triangle, taken as far down as is possible.
The green ones have a remainder of 1.
The purple, a remainder of 2, etc. You can follow the remainder colors and see the pattern.