Note: The Olympic Games were not held in 1940 and 1944.

Using technology, plot the data points on a graph. Define all variables used and state any parameters clearly. Discuss any possible constraints of the task.

What type of function models the behavior of the graph? Explain why you chose this function. Analytically create an equation to model the data in the above table.

On a new set of axes, draw your model function and the original graph. Comment on any differences. Discuss the limitations of your model. Refine your model if necessary.

Use technology to find another function that models the data. On a new set of axes, draw both your model functions. Comment on any differences.

Had the Games been held in 1940 and 1944, estimate what the winning heights would have been and justify your answers.

Use your model(s) to predict the winning height in 1984 and 2016. Comment on your answers.

The following table gives the winning heights for all the other Olympic Games since 1896.

...
IBMathSL
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...

...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...

...Alma Guadalupe Luna
MathIA (SL TYPE1)
Circles
Circles
Introduction
The objective of this task is to explore the relationship between the positions of points within circles that intersect.
The first figure illustrates circle C1 with radius r, centre O, and any point P. r is the distance between the centre O and any point (such as A) of circle C1.
Figure 1
The second diagram shows circle C2 with radius OP and centre P, as well as circle C3 with radius r and centre A. An intersection between C1 and C2 is marked by point A. The intersection of C3 with OP is marked by point P’.
Figure 2
Through this investigation I will examine how the r values correlate with the values of OP in determining the length of OP’ when r is held as a constant variable and the value of OP is the variable that is subject to change. I will then venture on to study the inverse, the relationship when the r values becomes the variable that is changed and the OP value is held constant.
r as a Constant
If we let the value of r be equal to 1, we can use that information to find the length of OP’ when OP=2, 3, and 4. The first thing one can deduce is that by using the points A, O, P’, and P two isosceles triangles can be formed; ∆AOP and ∆AOP’. To rationalize this assertion through an analytic approach it...

...Evaluate/Compare and Contrast/Discuss/Examine models or theories of one cognitive process with reference to research studies (22)
Human beings actively process information and it is cognitive processes that guide behavior. These cognitive processes are influenced by social and cultural factors. One of the cognitive processes is memory. Many researchers and psychologies have proved that the mind can be studies scientifically by developing theories and using a number of scientific research methods. This is demonstrated in theories and models of cognition which are discussed and continuously tested. The key characteristics of a good memory models are first, testability; second, simplicity; lastly, applicability. This essay looks at strength and limitation/similarities and differences/examine/discuss of models of one cognitive process, which is memory, with reference to research studies.
Not all of developing models have such characteristics above.
MSM by Atkinson and Shiffrin (1968)
* Sensory memory, STM and LTM as permanent structural components of the memory system.
* Rehearsal is a control process, which acts as permanent structural components of the memory system.
* Rehearsal is a control process, which acts as a buffer between sensory memory and LTM, and helps the transfer of information to LTM.
-Evidence
* H.M. Case studies by Milner (1966)
-Studies of brain-damaged, amnesic patients appear to support the STM-LTM distinction. While STM...

...curves mentioned above in one graph as follows:
From the graph, it can be concluded that the new function can fit the data, and better than the first function, but there is still some error between these two curves. New solution method or model should be advanced to describe tolerance of human beings to G-forces over time.
Then, we still found that some little error exist between real data and the function we got. Observe the time of the data, it is clear that there are large difference between the data, the scale of time is from 0.01 to 30, which is that uniformly distribute. So, logarithm axis is considered to scale the time. We can get new data table as follows:
Time (min) | Log Time (min) | Ln Time (min) | +Gx (g) |
0.01 | -2 | -4.61 | 35 |
0.03 | -1.52 | -3.51 | 28 |
0.1 | -1 | -2.3 | 20 |
0.3 | -0.52 | -1.2 | 15 |
1 | 0 | 0 | 11 |
3 | 0.478 | 1.1 | 9 |
10 | 1 | 2.3 | 6 |
30 | 1.478 | 3.4 | 4.5 |
Then we plot the graph of Log Time and Ln Time as follows:
From the graph, it can be seen that the shape of curve is similar with liner function. We assume the function form as follows:
So, we still use the new data to solve parameter k and b, we choose first and fifth group of data, second and the sixth group of data, third and seventh group of data, fourth and eighth group of data to solve parameters separately. Then we get the average of them as the final parameters.
To get k and b, I use the method of solving matrix...

...
Laurie Scott
SLMath Internal Assessment
Mr. Winningham
9/5/12
Instructions: In this task you will consider a set of numbers that are presented in a symmetrical pattern.
Pascal’s Triangle
|n=0 |1 |
|1 |0 |
|2 |3 |
|3 |6 |
|4 |10 |
|5 |15 |
|6 |21 |
Table 1: Relationship between Row Number and Numerator of Figure 2
[pic]
Figure 3: Graph of the relationship between Row Number and Numerator of Figure 2
In order to find the sixth and seventh rows, a pattern for determining the denominator must be found:
First it is helpful to determine a relationship between the numerator and denominator of the first term in each row:
|Row Number ( n ) |Difference of Numerator and |
| |Denominator (1st term) |
|1 |0 |
|2 |1 |
|3 |2 |
|4 |3 |
|5 |4 |
Table 2: Relationship between Row Number and the...

...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 slightly more difficult because there is more than one variable changing for each number of Lacsap’s fractions. All of the numbers in a row had the same numerator, but the denominator changes depending on which row the number is, and which term in the row it is.
Example: 1/1, 10/7, 10/6, 10/7, 1/1
In this example all of the numerators are 10 (1/1 can also be written as 10/10), but the denominators change.
So, to find the equation of the denominator, a relation must be set up between the denominator, D, and the term number, r, instead of the Numerator and the row number. However, because each row is unique, each row will have its own quadratic, therefore, a relation must be set up for each row.
Ex graph and data table:
Term # | D |
0 |...

...SLTYPE 1-LACSAP’S FRACTIONS
* INTRODUCTION
This investigation is going to do research patterns relates to the Lacsap’s Fractions. For its external structure, Lacsap’s Fraction is analogous to Pascal’s Triangle. Lacsap’s Fraction presents the way of generating and organizing the binomial coefficients. Within this investigation, the work is planning to be divided into two parts. In the first part, the content will relate to the pattern of numerators. In the second part, I am going to do the research on the patterns of denominator and the general statement for. Admittedly, the technology of computing will be involved into this investigation (E.g. Geogebro and GSP5chs). The following figure 1-1 illustrates Lacsap’s Fraction.
Fig.1-1
* PART A - CALCULATIONS and ANALYSIS
Firstly, I am going to research the numerator patterns. By observing the numerators of these fractions, it is illustrated that the first row of numerator is 1, second row of numerator is 3, third row of numerator is 6, fourth row of numerator is 10 and fifth row of numerator is 15. Let’s present it into the mathematical way: (= numerator of the row)
Continued
I realize that , , , Thus it is easily to find the numerator of the sixth row which is getting from. In order to do further investigation, it is essential to make a data table.
| numerator |
1 | 1 |
2 | 3 |
3 | 6 |
4 | 10 |
5 | 15 |
6 | 21 |
… | … |
n | ? |
Table...

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