- Level: International Baccalaureate
- Subject: Maths
- Word count: 2914
Lascap's Fraction Portfolio
Extracts from this document...
Introduction
MATHEMATICS SL PORTFOLIO
DZHARIF ALIAH BT SABRI
M11P
INTRODUCTION
The given triangle shows number that is arranged symmetrically from one row to another. Besides that, a lot of patterns can be derived from the existing elements making it possible to explore the other sub-elements within this triangle like the numerator and denominator.
Keys :
- n = row
- r = column
- Finding the numerator of the sixth row
A triangle consisting of only the numerators is made.
1 1
1 3 1
1 6 6 1
1 10 10 10 1
1 15 15 15 15 1
From the figure,
With the exception of the first and last columns which the value is fixed to ‘1’, a pattern can be observed from the other numerators. The difference between the value of the numerator from each row, n, to another row increases by 1.
r = 1 | r = 2 | ||||
Row (n) | Numerator (N) | Difference (Nn – N(n-1)) | Row (n) | Numerator (N) | Difference (Nn – N(n-1)) |
2 | 3 | 3 | 6 | ||
3 | 6 | +3 | 4 | 10 | +4 |
4 | 10 | +4 | 5 | 15 | +5 |
5 | 15 | +5 | 6 | 21 | +6 |
6 | 21 | +6 |
The highlighted columns show that the difference of the numerator between the rows increases by 1. The table above also shows an attempt to find the numerator when n = 6 which follows the pattern from the previous rows. Since the value of the numerator from each row, n, shows the same value, the numerator of the sixth row can be calculated by adding 6 to the numerator in the fifth row; 15. The numerator of the sixth row is as follows:
1 21 21 21 21 21 1
- Using technology, plot the relation between row number, n, and the numerator of each row. Describe what you notice from your plot and write a general statement to represent this.
Middle
7
22
+6
Key :
- n = Number of row
- D = Value of denominator
The table above shows the proof of the increment by 1 in the difference between denominators from different rows. Besides that, an attempt to predict the value of the denominator for the sixth and seventh row is also shown. Since the triangle is symmetrical, the value of the denominator for the sixth row is:
1 16 13 12 13 16 1
Following the symmetrical pattern from the previous rows of the triangle, the value of the denominator for the seventh row is:
1 22 18 16 16 18 22 1
Thus, the elements (E) of the sixth and seventh rows are:
n = 6 1 1
n = 7 1 1
- Find the general statement for En(r).
Explanation on the general statement.
En(r) is the element from nth row and rth column. Every element in the given triangle is in a fraction form and thus, each element is consist of a numerator (N) and a denominator (D) in the form of for the respective row and column. Thus, the general equation of En(r) must consists of . The general equation to find the numerator is already discovered in the early part of this portfolio where the general equation is:
N =
Finding the general equation of the denominator.
The values of elements in the triangle all gives us clues on the patterns and correlations between one another. The difference of the numerator and denominator for elements in the same row also forms a pattern. Instead of finding the difference of denominators between rows, denominators can also be calculated by substracting the value of the difference between the numerator and denominator (N – D) from the value of numerator (N). The table belows gives a clearer picture of the claim made in the earlier sentence.
n = 6 | n = 7 | ||||||||
r | N | (N – D) | N – (N – D) | D | r | N | (N – D) | N – (N – D) | D |
1 | 21 | 5 | 16 | 16 | 1 | 28 | 6 | 22 | 22 |
2 | 21 | 8 | 13 | 13 | 2 | 28 | 10 | 18 | 18 |
3 | 21 | 9 | 12 | 12 | 3 | 28 | 12 | 16 | 16 |
Equal values
Equal values
The table above proves that the value of denominator can be derived from subtracting the difference of the numerator and denominator (N – D) from the numerator (N);
D = N – (N – D)
Since the general equation of N is known, now, I only need to find any correlation between the row number, column number and (N – D). This is to make it easier to formulate the general equation of the denominator. After a few trials, the value of (N – D) is actually the result of multiplying the difference between n and r with r, r(n – r). Table below gives a clearer picture of the newly derived equation for (N – D).
6th row | 7th row | ||||||||
n | r | (n – r) | r(n – r) | (N – D) | n | r | (n – r) | r(n – r) | (N – D) |
6 | 1 | 5 | 5 | 5 | 7 | 1 | 6 | 6 | 6 |
6 | 2 | 4 | 8 | 8 | 7 | 2 | 5 | 10 | 10 |
6 | 3 | 3 | 9 | 9 | 7 | 3 | 4 | 12 | 12 |
Conclusion
- The general statement is not applicable for row n = 0. Looking at the general statement, [, if value n = 0 is plugged into the equation, the solution will be 0 based on the rule that states that, any values that is multiplied by 0 equals to 0. Besides that, when the value of n = 0, the numerator will always be 0 and in the end will always give 0 as the answer. Thus, this general statement will not provide the element for n = 0, r = 0.
Example:
When n = 0, r = 1
E0(1) = [
=
= 0
- Since the pattern of the elements form a triangle which starts with column n = 1, the value of n < 0 will be undefined. Thus, plugging in negative values of n will no longer follow the original triangular shape.
- The general stament is also is limited to column r0. Any negative values for r will no longer match the symmetrical pattern of the triangle.
CONCLUSION
The regular patterns shown throughout the triangle allows us to figure out the upcoming rows of the triangle. General equations of the numerator and the denominator are able to derived and thus, makes it easy to calculate the elements in other rows and columns. However, there are still limitations to this general statement. Since the shape that the symmetrical elements form is a triangle, the range of elements that can be calculated is limited to within the shape of the triangle only. For example, catering for n < 0 will no longer follows the triangle shape. All in all, observing patterns that the fractions potray enables us to discover what lies beneath all the numbers.
[1]http://calculator.maconstate.edu/quad_regression/index.html
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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