In Lacsap's Fractions, when looking for a general pattern for the numerator, it can be noted that it does not increase linearly but exponentially. Numerators are 3,6,10, and 15, each preceding numerator added by one plus the row number. Using this general statement it can be concluded that the numerator in the 6th row is 21 (15+6), and 28 for the 7th.

Generating a Statement for the Numerator:
To generate an equation for the numerator of the fraction, the fraction data must be organized and graphed. The table below shows the relationship between the row number and numerator being relative to an exponential function as the sequence goes on. N(n+1)-Nn represents the equation for the graph that increases more evenly as the sequence advances.

Using excel to graph the points and loggerpro to generate an equation, the general statement for finding the numerator N=0.5n2+0.5n, n having to be greater than 0. To check the validity of the equation sample equations were used: Sample Equation:

5th Row: N=0.5(5)2+0.5(5)=15
Patterns Recognized:
The first pattern that could be recognized is that the difference between the numerators of the ensuing rows is 1 more than the change between the previous numerator of the two consecutive rows. The formula that represents the pattern of how to find the numerator is N(n+1)-N(n)=N(n)-N(n-1)+1. Using this method, the 6th and 7th rows can be found:

This is only a supplement to the equation found in the graph above (N=0.5n2+0.5n). This pattern only tests the validity of the equation derived from the table because of both methods concluding to the same value.

Generating a Statement for the Denominator:
To examine the denominators in Lascap's Fractions, the values for the 6th row and their corresponding elements were put onto a table, and ultimately a graph. Showing a pattern, it...

...In Lacsap’s Fractions, En(r) refers to the (r+1)th term in the nth row. The numerator and denominator are found separately, therefore to find the general statement, two different equations, one for the numerator and one for the denominator, must be found. Let M=numerator and let D=denominator so that En(r) = M/D.
To find the numerator for any number of Lacsap’s Fractions, an equation must be made that uses the row number to find the numerator. Because the numerator changes depending on the row, the two variables (row number and numerator) must be compared. To find this equation, the relationship between the row number and numerator must be found, put it graph form, and the equation for the graph will be the equation needed.
Row Number, n | Numerator, N |
1 | 1 |
2 | 3 |
3 | 6 |
4 | 10 |
5 | 15 |
Numerator
Numerator
Row Number
Row Number
The equation for the numerator can be derived by using quadratic regression on a graphing calculator. The equation is; y = .5x2 + .5x. This translates into; M=.5n2+.5n, where n=row number, and M=numerator. This means that any numerator from a certain row number can be found by using this equation. For example, to find the numerator of the sixth row, “6” needs to be substituted in for n.
M= .5n2 + .5n
M= .5(6)2 + .5(6)
M= .5(36) + .5(6)
M= 18 + 3
M= 21
The Numerator for row six is 21
They method to find the equation for the denominator is similar, but...

...Jonghyun Choe
March 25 2011
MathIBSL
Internal Assessment – LASCAP’S Fraction
The goal of this task is to consider a set of fractions which are presented in a symmetrical, recurring sequence, and to find a general statement for the pattern.
The presented pattern is:
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
Step 1: This pattern is known as Lascap’s Fractions. En(r) will be used to represent the values involved in the pattern. r represents the element number, starting at r=0, and n represents the row number starting at n=1. So for instance, E52=159, the second element on the fifth row. Additionally, N will represent the value of the numerator and D value of the denominator.
To begin with, it is clear that in order to obtain a general statement for the pattern, two different statements will be needed to combine to form one final statement. This means...

...IBMathsSL TYPE I
Lacsap’s Fractions Portfolio
Lacsap’s fraction
Introduction:
Lacsap’s fraction is a symmetrical triangle that has the following pattern in the first five rows
The shape is similar to Pascal triangle. It has the same quantity of symmetrical triangle as Pascal triangle. And Lacsap is the inverse alphabet order of Pascal. These make me think about Pascal triangle and I made an assumption that elements in Lacsap triangle may have the same relationship as in Pascal triangle. However, the elements in Lacsap’s fraction triangle may or may not have the same relationship as Pascal triangle so I ignore my hypothesis about Pascal triangle and decide to find the relationship by not referring to Pascal triangle. I believe that every element in Lacsap’s triangle must be in a sequence and the task for this portfolio is to find the relationship between each element.
This portfolio will be divided into five parts; finding numerator relationship, finding denominator relationship, test the accuracy of the statement, finding additional rows and limitations of the statement.
The notations in this portfolio are
* X = the element place
* N = row number
* En(x) = The xthelement on the nthrow
Numerator Relationship:
First thing I notice about the Lacsap is that the numerators are the same in the same row.
The numerators are the same in each row...

...Lacsap’s Fractions
The aim if this IA is to investigate Lacsap’s Fractions and to come up with a general statement for finding the terms.
When I noticed that Lacsap was Pascal spelt backwards I decided to look for a connection with Pascal’s triangle.
Pascal’s triangle is used to show the numbers of ‘n’ choose ‘r’(nCr). The row number represents the value of ‘and the column number represents the ‘r’ value. Eg. Row 3, colomn 2 = 3C2 = 2.
I noticed that all the numerators of the fractions in Lascap’s fraction (3,6,10,15) are also found in Pascal’s triangle. So I tried to see if I would get the denominator of the fractions by using the row as ‘n’ and the colomn (or element) as ‘r’. This did not work out because Lascap’s triangle does not have a row with only one element like Pascal’s does. To solve this I just added 1 to each row number. This gives me the formula[pic].
|(Row number +1)C2 |Numerator |
|(2+1)C2 |= 3 |
|(3+1) C2 |= 6 |
|(4+1)C2 |=10 |
|(5+1)C2 |=15 |
Now that we have found an equation to solve to numerator of the...

...IB Mathematics SL Year 1
Welcome to IB Mathematics. This two-year course is designed for students who have a strong foundation in basic mathematical concepts. The topics covered in this course include:
* Algebra
* Functions
* Equations
* Circular functions
* Trigonometry
* Vectors
* Statistics
* Probability
* Calculus
-------------------------------------------------
Resources:
* Textbook: Mathematics SL 3rd edition. Haese Mathematics 2012
ISBN: 978-1-921972-08-9
* Edmodo: A virtual learning website where students can get homework assignments, ask homework questions, access class documents, and communicate with peers. https://iam.edmodo.com/
* Class code: ybu2s4
* Quest: Virtual learning website where students will complete graded homework assignments. https://quest.cns.utexas.edu/
* Graphing Calculator. Recommended for this course and all subsequent math courses. The Ti-Nspire or Ti-84 plus silver edition are both good choices. http://education.ti.com
* Khan Academy: http://www.khanacademy.org. Will be used to provide students with alternative resources.
* Geometer’s Sketchpad: Graphing software. Will be installed on student computers. http://www.dynamicgeometry.com/
* Additional print, online resources and worksheets may be used.
-------------------------------------------------
Assignments:
There...

...Lacsap’s FractionsIBMathSL
Internal Assessment Paper 1
Lacsap’s Fractions
Lacsap is Pascal spelled backward. Therefore, Pascal’s Triangle can be used practically especially with this diagram.
(Diagram 1)
This diagram is of Pascal’s Triangle and shows the relationship of the row number, n, and the diagonal columns, r. This is evident in Lacsap’s Fractions as well, and can be used to help understand some of the following questions.
Solutions
Describe how to find the numerator of the sixth row.
There are multiple methods for finding the numerator of each consecutive row; one way is with the use of a formula, and another by using a diagonal method of counting illustrated by a diagram.
The following image can be used to demonstrate both techniques to finding the numerator:
(Diagram 2)
This formula uses “n” as the row number and the outcome is the numerator of the requested sixth row.
n2 + n
2
As indicated, inputting the number 6, as the requested sixth row, for n gives the solution of 21.
X = n2 + n
2
X = (6)2 + (6)
2
X = 36 + 6
2
X = 42
2
X = 21
Therefore, as shown, the numerator of the sixth row is 21, and this can be checked for validity by entering each number, 1 through 5, into the formula and making sure that the answer corresponds with the numerator in the above diagram.
Where n = 5:...

...Exploration of Lacsap’s Fractions
The following will be an investigation of Lacsap’s Fractions, that is, a set of numbers that are presented in a symmetrical pattern. It is an interesting point that ‘Lacsap’ is ‘Pascal’ backwards, which hints that the triangle below will be similar to “Pascal’s Triangle”.
1 1
1 1
1 1
1 1
1 1
There are many patterns evident in this triangle, for instance I can see that there is a vertical axis of symmetry down the middle of the triangle. Each row starts and ends with the number 1. Each row has one more variable than the number of rows, i.e. row 1 has 2 variables. The numerators in the middle stay the same and the diagonals form sequences.
In order to decipher the pattern in the numerators and denominators, I had to look at the triangle a different way. Knowing that the numerators of the row don’t change, it occurred to me that the number 1s on the outside of the triangle could be expressed as fractions.
This proves that all the numerators of the row are the same.
To further investigate the numerators, I will examine the relationship between the
row number and the numerator, which is shown in the table below. These are the numerators after having changed the 1s on the outside of...

...MathSL Portfolio – Tips and Reminders Checklist
Notation and Terminology
Check for the following:
• I did not use calculator notation. (I didn’t include things like ‘x^2’ for or Sn for Sn)
• I used appropriate mathematical vocabulary.
Communication
Check for the following:
• The reader will not need to refer to the list of questions in order to understand my work.
• My responses are not numbered.
• I have an introduction, conclusion, title page, and table of contents.
• All graphs are labeled – Each graph has a title, labeled axes, and appropriate scale.
• My graphs and tables are within the body of my work. They are not separate or in an appendix.
• I have explained why I made the choices I did when going through the task.
• I did not include key stroke sequences, e.g. “I pressed the 2nd key, then TRACE…”
• My tables do not straddle pages.
• My tables are labeled well, including my variable definitions in each column.
Use of Technology
Check for the following:
• I used technology to illustrate my points and ideas. I didn’t just “stick in” a graph.
• Each graph or table (or other piece of tech.) is accompanied by explanations and my ideas.
• I did not include too many graphs on the same axes – my graphs are easy to read.
Mathematical Process (Type 1)
Check for the following:
• I explicitly defined variables and parameters the first time I used them, even if they were
already defined in...