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Pascal's Triangle of Doom
Apr. 7th, 2010 10:32 pmI'm sure there are more than a few people on my friends list who think Pascal's Triangle spells doom without adding anything to the mix, but bear with me.
My student teacher and I took a grade four activity about probabilities in coin tosses (what is the likelihood of throwing two heads on two coins? How about three heads on three coins? Two heads and one tail on three coins?) The probabilities with coin tosses just happen to fit in beautifully with Pascal's Triangle, one of the more famous (and complex) number patterns out there. The National Library of Virtual Manipulatives" has an excellent starting point a ways down that page.
Well, it occurred to me that this might make an interesting quilt pattern. Could I choose colours that would reflect certain multiples and go from there? I realized pretty quickly tht making it in the shape of a triangle was at best problematic, but I could make it more like fraction strips (which I'm sure you can find if you poke about on that website a bit) so it would be a rectangle cut into fractions to represent the numbers in the pattern.
Well, so far, so good. The problem came when I realized I didn't just want to divide the strip in half, then in quarters, then in eighths, with the numbers of the sequence in each piece. What I wanted to was to represent the numbers themselves as part of the fractions. So, the zero line is one whole strip; the first line has two ones, which can be represented as halves of the strip; but the second line is 1, 3, 3, 1. I don't want to represent that as quarters; I want to represent it as eighths, where the two threes in the middle are three times larger than the ones at the side and set apart from each other using colour or texture. The next line after that is 1, 4, 6, 4, 1; that's a total of sixteen. The ones can remain the same colour as the other ones to give the impression of a triangle as I descend the quilt, and I kind of want to bring multiples into it, so the six would have to be a similar colour scheme to the two threes in the line above, and the two fours would have to be totally different.
It then occurred to me that the possibility of pulling colour theory into this exists in primary, secondary, and tertiary colours (though it gets a bit hazy after that and may need to be accomplished with texture and multiples of those colours.) So, if one is black, I could start with red for multiples of three and blue for multiples of two. So the zero strip would be black; the first would also be black but halved using texture somehow; the second would be black, redredred, a different redredred, black; the third would be black, royal blue in a four square, purple (for 2x3) in six parts, blue again in a four square, and black.
The next row has another prime number, five, which then needs to be - yellow, I guess, to use up the primary colours? - then there are tens in there so yellow + blue = green. The next row has six again, so purple, followed by fifteen, which is 5x3 to yellow + red = orange, then twenty which brings in gray for the first time, multiplying 2x3x5, or all three primary colours. Or I could skip the primary colours for five and go straight to a tertiary colour to make it more interesting.
My head's starting to hurt but I just CAN'T STOP.
Can anyone else follow this at all?
My student teacher and I took a grade four activity about probabilities in coin tosses (what is the likelihood of throwing two heads on two coins? How about three heads on three coins? Two heads and one tail on three coins?) The probabilities with coin tosses just happen to fit in beautifully with Pascal's Triangle, one of the more famous (and complex) number patterns out there. The National Library of Virtual Manipulatives" has an excellent starting point a ways down that page.
Well, it occurred to me that this might make an interesting quilt pattern. Could I choose colours that would reflect certain multiples and go from there? I realized pretty quickly tht making it in the shape of a triangle was at best problematic, but I could make it more like fraction strips (which I'm sure you can find if you poke about on that website a bit) so it would be a rectangle cut into fractions to represent the numbers in the pattern.
Well, so far, so good. The problem came when I realized I didn't just want to divide the strip in half, then in quarters, then in eighths, with the numbers of the sequence in each piece. What I wanted to was to represent the numbers themselves as part of the fractions. So, the zero line is one whole strip; the first line has two ones, which can be represented as halves of the strip; but the second line is 1, 3, 3, 1. I don't want to represent that as quarters; I want to represent it as eighths, where the two threes in the middle are three times larger than the ones at the side and set apart from each other using colour or texture. The next line after that is 1, 4, 6, 4, 1; that's a total of sixteen. The ones can remain the same colour as the other ones to give the impression of a triangle as I descend the quilt, and I kind of want to bring multiples into it, so the six would have to be a similar colour scheme to the two threes in the line above, and the two fours would have to be totally different.
It then occurred to me that the possibility of pulling colour theory into this exists in primary, secondary, and tertiary colours (though it gets a bit hazy after that and may need to be accomplished with texture and multiples of those colours.) So, if one is black, I could start with red for multiples of three and blue for multiples of two. So the zero strip would be black; the first would also be black but halved using texture somehow; the second would be black, redredred, a different redredred, black; the third would be black, royal blue in a four square, purple (for 2x3) in six parts, blue again in a four square, and black.
The next row has another prime number, five, which then needs to be - yellow, I guess, to use up the primary colours? - then there are tens in there so yellow + blue = green. The next row has six again, so purple, followed by fifteen, which is 5x3 to yellow + red = orange, then twenty which brings in gray for the first time, multiplying 2x3x5, or all three primary colours. Or I could skip the primary colours for five and go straight to a tertiary colour to make it more interesting.
My head's starting to hurt but I just CAN'T STOP.
Can anyone else follow this at all?
