Ib Math Project 1
Lacsap’s Fractions
IB Math SL
SL Type 1
December 11, 2012 Lacsap’s Fractions: Lacsap is Pascal backwards and the way that Lacsap’s fractions are presented is fairly similar to Pascal’s triangle. Thus, various aspects of Pascal’s triangle can be applied in Lacsap’s fraction. To determine the numerators: To determine the numerator (n), consider it in relation to the number of the row (r) that it is a part of. Consider the five rows below: Row 1 1 1 Row 2 1 32 1 Row 3 1 64 64 1 Row 4 1 107 106 107 1 Row 5 1 1511 159 159 1511 1 The relation between the numerator and the row number can be shown by the equation: Where the numerator = n And the row = r 0.5r2 + 0.5r = the numerator (n) When r=1 0.5(12) + 0.5(1) = 1 When r=2 0.5(22) + 0.5(2) = 3 When r=3 0.5(32) + 0.5(3) =6 When r=4 0.5(42) + 0.5(4) = 10 When r=5 0.5(52) + 0.5(5) = 15 Therefore if you are attempting to find the 6th or 7th row numerators, you simply plug (n) into the equation: When r=6 0.5(62) + 0.5(6) = 21 When r=7 0.5(72) + 0.5(7) = 28 Therefore the terms in the 6th row include: 1 2116 2113 2112 2113 2116 1 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 | 105 | 120 | 136 | 153 | 171 | 190 | 210 | Thus, the terms in the 7th row include: 1 2822 2818 2816 2816 2818 2822 1 The following graph and chart show the relation between the number of the row, and the numerator:
There is a definite pattern in this portion of the fraction: The number of the numerator increases by one more than the last numerator was raised by. Finding the
IB Math SL
SL Type 1
December 11, 2012 Lacsap’s Fractions: Lacsap is Pascal backwards and the way that Lacsap’s fractions are presented is fairly similar to Pascal’s triangle. Thus, various aspects of Pascal’s triangle can be applied in Lacsap’s fraction. To determine the numerators: To determine the numerator (n), consider it in relation to the number of the row (r) that it is a part of. Consider the five rows below: Row 1 1 1 Row 2 1 32 1 Row 3 1 64 64 1 Row 4 1 107 106 107 1 Row 5 1 1511 159 159 1511 1 The relation between the numerator and the row number can be shown by the equation: Where the numerator = n And the row = r 0.5r2 + 0.5r = the numerator (n) When r=1 0.5(12) + 0.5(1) = 1 When r=2 0.5(22) + 0.5(2) = 3 When r=3 0.5(32) + 0.5(3) =6 When r=4 0.5(42) + 0.5(4) = 10 When r=5 0.5(52) + 0.5(5) = 15 Therefore if you are attempting to find the 6th or 7th row numerators, you simply plug (n) into the equation: When r=6 0.5(62) + 0.5(6) = 21 When r=7 0.5(72) + 0.5(7) = 28 Therefore the terms in the 6th row include: 1 2116 2113 2112 2113 2116 1 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 | 105 | 120 | 136 | 153 | 171 | 190 | 210 | Thus, the terms in the 7th row include: 1 2822 2818 2816 2816 2818 2822 1 The following graph and chart show the relation between the number of the row, and the numerator:
There is a definite pattern in this portion of the fraction: The number of the numerator increases by one more than the last numerator was raised by. Finding the