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
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
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
Section 9.2
a) Let x represent the number of loads of laundry each sister use per week
Let y represent liters of water Sharon uses to wash her car each time
Sharon
225x
y
225x+y
Laundry
Car wash
Total water usage
Bev
95x
y+260
95x+y+260
225x+y=95x+y+260
b
Interpolation and Extrapolation
Had the Games been held in 1940 and 1944, estimate what the winning heights would have been
and justify your answers.
Year 1940 = Year # 2
Verify:
Year 1944 = Year # 3
Verify:
After adjustments, the Base-10 Logarithm function on this new set of axis is:
Formula:
Variables:
y: Heights(cm)
x: Years
: Vertical stretch by a factor of
: Horizontal stretch by a factor of
: Horizontal displacement 18.59 units to the left
Above paramet
Another function that models that data: Sinusoidal Function
Formula:
Variables:
y: Heights(cm)
x: Years
: amplitude of this function is 19.21
period of this function is
: phase shift of to the left
: Vertical displacement of 217.9 units up
Above parameter
If we apply the previously calculated Base-10 Logarithm function,
to this new set of combined data, we will see that it no longer fits because the x-values of that
function has been shifted.
RMSE=26.769cm
The most significant fluctuation is from year#20 (
Use your model you predict the winning height in 1984 and in 2016. Comment on your answers.
Year 1984 = Year # 13
Although the predicted height in year 1984 is a little lower than the height achieved in 1980,
233.81cm is a fairly reasonable prediction.
Ye
Base-10 Logarithm Fit:
Zoomed-out graph:
Formula:
Variables:
y: Heights(cm)
: Vertical stretch by a factor of 3398
x: Years
: Horizontal stretch by a factor of
Above parameters are all with respect to the equation,
RMSE: 4.550cm
From the zoomed out graph
Linear Fit:
Formula:
Variables:
y: Heights(cm)
x: Years
Slope:
y-intercept:
RMSE: 4.523cm
Root Mean Square Error (RMSE) is when you take the square root of the sum of all differences
between the estimated y-value and the actual y-value of the data. The lo
Data Analysis
The table below gives the height (in centimeters) achieved by the gold medalists at various
Olympic Games.
Year
Height(cm
)
1932
197
1936
203
1948
198
1952
204
1956
212
1960
216
1964
218
1968
224
1972
223
1976
225
1980
236
Note: The Olympic
After comparing the above three graphs, Linear Fit, Sine Fit, and Base-10 Logarithm Fit, we can
see that the RMSE value of the Sine Fit is the lowest. Theoretically, this would be the best
function to model the set of data we are provided with, and the in