Spiralaterals POW
For this Problem of the Week, we investigated spiralaterals. We first chose a sequence of numbers. We could change a few different variables in the numbers, including changing the number of numbers, the number itself, changing it from positive to negative and vice versa, etc. You could create patterns with the numbers, using only odd or even numbers, counting up or down from a certain number, and others. These are only a few things I explored in this problem. After choosing your numbers, you begin to draw your spiralateral. You pick a point (using graph paper is easiest) and go up as many spaces as your first number. Then you go to the right the number of spaces as your second number, then down, then to the left, and finally back up. You can continue your spiral as long as you want, sometimes it eventually just completes the circuit and it will continue the same path again. The question I mainly focused on during this problem was even number sequences and odd number sequences.
Beginning this problem, I played around with many different variables. Number of number in the sequence, switching numbers around, positive and negatives, and finally even and odd number sequences. After drawing many different spiralaterals, I saw one of the most interesting changes when going between odd and even numbers. Below are a few of the sequences I tried, both odd and even. I tried 1,3,5,7,9, and 2,4,6,8,10. Comparing the two, the even number sequence is a bigger spiralateral even though there are the same number of numbers in the sequence. I also tried 1,3,5 and 2,4,6. I noticed the same thing in this comparison as well. The spiralaterals didn’t change direction, and continued clockwise, unlike the sequences where you counted down backwards instead of forwards. The spiral was also always completed. I didn’t see any other major changes besides those.
Beginning this problem, I played around with many different variables. Number of number in the sequence, switching numbers around, positive and negatives, and finally even and odd number sequences. After drawing many different spiralaterals, I saw one of the most interesting changes when going between odd and even numbers. Below are a few of the sequences I tried, both odd and even. I tried 1,3,5,7,9, and 2,4,6,8,10. Comparing the two, the even number sequence is a bigger spiralateral even though there are the same number of numbers in the sequence. I also tried 1,3,5 and 2,4,6. I noticed the same thing in this comparison as well. The spiralaterals didn’t change direction, and continued clockwise, unlike the sequences where you counted down backwards instead of forwards. The spiral was also always completed. I didn’t see any other major changes besides those.
The only patterns I came up with were that the even numbers create bigger spiralaterals than the odd numbers, even if there are the same number of numbers in the sequence. I tried a few different numbers and it always came out with that same conclusion. The spirals also continued in a clockwise direction instead of counter clockwise even with different number sequences. I don’t know what other solution I was supposed to come up with other than that.
I don’t really know what I learned from this problem, as I feel the spiral problem we did last year was a lot more advanced than this, and I still didn’t learn that much. The two habits of a mathematician that I used for this problem were being patient, because I had to do a lot of tedious spiral drawings of the same thing over and over again. The second habit of a mathematician that I used was looking for patterns. In order to grasp something from this problem, I had to a lot of comparisons between spiralaterals to find out why certain sequences turned out the way they did.
I don’t really know what I learned from this problem, as I feel the spiral problem we did last year was a lot more advanced than this, and I still didn’t learn that much. The two habits of a mathematician that I used for this problem were being patient, because I had to do a lot of tedious spiral drawings of the same thing over and over again. The second habit of a mathematician that I used was looking for patterns. In order to grasp something from this problem, I had to a lot of comparisons between spiralaterals to find out why certain sequences turned out the way they did.