The solution to this system of equations result in the values A= , B=, C= 0. Therefore, the
quadratic equation is, or it can also be rewritten as . By replacing the variable x with the variable
n, representing the row number, and y with the numerator, the
In this assignment, students are required to find patterns within a triangle of elements, which
look very similar to Pascals triangle. Pascals triangle is a triangle of numbers, commonly used
for binomial expansions.
The variable, r, will represent the nu
first and last term must be the numerator over itself. When r= 0, the denominator is the same as
the numerator, which means that the y-intercept of the quadratic equation must be the
numerator.
In order to
find the equation to predict the denominator of a
The quadratic equations shown in Figure 1 show that for each of the first seven rows:
And when y is replaced by the variable n, and x is replaced by the variable r:
, where
In this case, substitute the equation for the Numerator into :
Now, you can factor
In this triangle, it is much more difficult to predict the denominator, since it changes between
rows, and in each position within each row. In order to determine the denominators for the sixth
and seventh row, I used the pattern shown in Figure 5. By pur
ARCHBISHOP MACDONALD HIGH SCHOOL
LACSAP'S FRACTIONS
Received Date: December 07, 2012
Due Date: December 20, 2012
Abstract
This document presents the results of studying Lacsap's Fractions. The results include (1) the
relation between the row number and th
2. Finding the Numerator:
1. The numerators in each row are: 3, 6, 10, 15
2. Insert the row number in the L1 column for the x values
3. Insert the numerators in the L2 column for the y values
2.1 Graph:
Since the row number and the numerator can only be p
The solution to this system of equations result in the values A= , B=, C= 10. Therefore, the
quadratic equation is. By replacing the variable x with the variable r, representing the number of
the element in the row, and y is replaced by the value of the e
4. Finding the General Statement:
4.1 General statement:
Let En(r) be the rth fraction in nth row.
For example: E4(2) = 10/6
Then, the general statement is:
En(r) =
4.2 Verify:
E5(3) =
E5(3) =
4.3 Calculations:
The 6th row (The number of fractions in the
1. Patterns:
1.1 Pattern 1:
1) The first and last element in nth row = 1
2) Fractions in nth row are symmetrical: E(i) = E(r-i), 1 i r/2
1.2 Pattern 2:
In each row, the number of elements is equal to the row number plus one.
r+2 = n+1, r and n = 1,2,3,.
F