Today while practice counting, my cooperating teacher and I decided on having students count by 11 and 37 as far as they could get in one minute on paper. 11's were predictable and the students saw the traditional patterns (numbers counting up, etc). The fun part comes when students were counting by 37, and the teachers in the room took on the challenge as well. We found that when the multiple of 37 were placed in a certain way, (ie. a 9 down by 3 across array), patterns began to emerge. Students were able to pick out a few, then with come guidance were able to figure out more complex ones.
New patterns emerged as multiples of 37 broke into 4 digit numbers, which held true to 5 digit numbers. With the patterns and equations students came up with, we were able to find the 24,000th multiple of 37 within 5 seconds using mental math only, then the 54,000th multiple, then the 81,000th multiple within the same time frame.
It has been a long time since I have seen a group of 5th graders so excited about math and willing to self explore and discover and go through trial and errors that stunned the principal.
Plus! Students felt so great about themselves, it was a great confidence booster to them all, no matter their achievement level, because they were all able to come up with atleast one pattern that worked for them.
Comment wityh what you find, or request some of my own student work and patterns!
EDIT:
Here is an example of student work. You can see their efforts to find some of the trickier hidden patters and algorithms.
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