Introduction and purpose of task: The purpose of this task is to investigate the positions of points in intersecting circles and to discover the various relationships between said circles. Circle C1 has center O and radius r. Circle C2 has center P and radius OP. Let A be one of the points of intersection of C1 and C2. Circle C3 has center A and radius r (therefore circles C1 and C3 are the same size). The point P’ (written P prime) is the intersection of C3 with OP. This is shown in the diagram below.

Analytically find OP’ using r=1 and OP=2, OP=3, and OP=4:
First, I created a line (see the dashed line in the above figure) between AP’ that creates the ΔAOP’. Because P’ is on the circumference of circle C3 and A is the center of circle C3, that means that AP’ is equal to the radius of C3, which is 1. We also know that because line AO connects the circumference of C1 with the center of C1 (O) and the circumference of C3 with the center of C3 (A), the radii of these circles is the same, which means that they are equivalent circles. Therefore, in the ΔAOP’, AO=AP. When a triangle has two equivalent sides, it is an isosceles triangle. By that logic, ∠O=∠P’. Now, I looked at the triangle that is already drawn in the above figure, ΔAOP. We know that this triangle is also isosceles because OP=AP. By that logic, ∠A=∠O.

Using the law of cosines c^2=a^2+b^2-2abcos(C), which works for any triangle, I assigned θ to ∠O and determined that cos(θ)=1/(2*OP). Then, using the law of sines (insert law of sines here), sin(θ)/1=sin(180-2θ)/OP’ OP’=sin(180-2θ)/sin(θ)

OP’=sin(2θ)/sin(θ)
OP’=2cos(θ)
But because cos(θ)=1/2OP as earlier discovered;
OP’=1/OP

By using this equation, I derived the following answers analytically using r=1 and OP=2, OP=3, and OP=4. OP234
OP'0.50.330.25

Behavior of intersecting circles and general statement describing interaction that occurs when value of OP is changed: As OP changes, the resulting OP’ value...

...Article Review 1
DeGeorge, B., Santoro, A. (2004). “Manipulatives: A Hands-On Approach to Math.” Principal, 84 (2), (28-28).
This article speaks about the importance and significance of the use of manipulatives in the classroom, specifically in the subject of math. Manipulatives have proven to be valuable when used in a math class and are even more valuable to the children when they are young, and are learning new math concepts. Students are able to physically visualize the math concepts and gain knowledge because they understand what they’re learning a whole lot better and they also are able to gain insights on those concepts. Different examples of manipulatives may include counting with beans or M&M’s, using pattern blocks, puzzles, tangrams, and flash cards, just to name a few.
Using manipulatives in a math class are beneficial to both the student and the teacher because the teacher is able to explain the concepts to the students in a much easier manner using the hands-on technique, rather than explaining it verbally. It’s especially beneficial to the student because by incorporating these manipulatives into their learning process, they are able to pick up the concepts much quicker and in a way that they better understand, yet are having fun while doing it. When they have the concepts down, the students’ self-esteem goes up and they feel encouraged to keep on going.
After...

...MathIAMath Internal Assessment
EF International Academy NY Student Name: Joo Hwan Kim Teacher: Ms. Gueye Date: March 16th 2012
Contents
Introduction Part A Part B Conclusion
Introduction
The aim of this IA is to find out the pattern of the equations with complex numbers by using our knowledge. I used de Moivre’s theorem and binomial expansion, to find out the specific pattern and make conjecture about it. I basically used property of binominal theory with the relationship between the length of the line segments and the roots.
Part A
To obtain the solutions to the equation ) | | Moivre’s theorem, (| | equation, we will get: , I used de Moivre’s theorem. According to de . So if we apply this theorem in to the
(| |
)
(
(| |
)
)
| |
(
)
If we rewrite the equation with the found value of , it shows (| | ( ( ( ( ) )) ))
Let k be 0, 1, and 2. When k is 0, ( ) ( )
√
√
Now I know that if I apply this equation with the roots of
( )
( ) we can
find the answers on the unit circle. I plotted these values in to the graphing software, GeoGebra and then I got a graph as below:
Figure 1 The roots of z-1=0 I chose a root of and I tried to find out the length of two segments from the point Z. I divided each triangle in to two same right angle triangles. By knowing that the radius of the unit circle is 1, with the knowledge of the length from D or Z to their...

...Name: Linh Nguyen
IB MathIA
02/06/12
Part A
Consider this 2× 2 system of linear equations: x + 2y = 3
2x - y = -4
We can see patterns in the constants of both equations. In the first equation, the constants are 1, 2, and 3. The common difference between the constants is 1:
3 – 2 = 1
2 – 1 = 1
Based on this, we can set up a general formula for the constant of this equation:
Un = U1 + (n - 1)d Where:
n: The number of the series
d: the common difference in the series.
For the second equation, we also can see that the constants belong to a arithmetic series, which has the common difference of -3:
-1 – 2 = -3
-4 – (-1) = -3
Un = U1 + (n+3)d
Solving the equation:
x + 2y = 3
2x – y = -4
* 2x + 4y = 6
2x – y = -4
* 5y = 10
* y = 2
* x + 4 = 3
* x = -1
Looking at the graph, we can see the intersection point A has the coordinate (-1, 2). At this point, the two lines are equal.
Solving equation with similar formats:
x + 6y = 11 Common difference is 5
6x – 2y = -10 Common difference is -8
* 6x + 36y = 66
* 38y = 76
* y = 2
* x = -1
6x + 7y = 8 Common difference is 1
2x + 4y = 6 Common difference is 2
* 6x + 12y = 18
* 5y = 10
* y = 2
* x = -1
Looking at these systems of linear equations, we can conclude that any equations which have their constants follow a arithmetic series will have the answer for the variable y as 2 and variable x as -1....

...product of their probabilities, and use this characterization to determine if they are independent.
3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model
6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the...

...IB Math Studies Internal Assessment:
Is the distance a tennis ball travels horizontally dependent on the angle of which it is dropped at?
Exam Session: May 2014
School Name:
Teacher:
Course: IB Math Studies
Word Count: 654
Name:
Is the distance a tennis ball travels horizontally dependent on the angle of which it is dropped at?
Introduction
In tennis, players hit the tennis ball in certain ways so the ball goes the way they want it to go. Hitting it at certain angles enables the tennis ball to travel various distances, it depending on the angle. Some angles make the ball go short or far distances.
Statement of Task
The main purpose of this investigation is to determine whether there is a relationship between the distance a tennis ball travels horizontally and the angles of which it is dropped at. The type of data that will be collected is that I will drop a tennis ball, at a constant height of 4 feet, at from different angles (30°, 40°, 50°, 60°, 70°), twenty times for each angle with a total of one-hundred drops and measuring the distance of each drop in feet. The data will be used to determine whether the angle of which a ball is dropped affects how far a tennis ball travels horizontally.
Plan of Investigation
I am investigating if the distance a tennis ball travels horizontally dependent on the angle it is drops at. I have collected data on the distances a tennis ball travels...

...MathIA
IB MATH SL
MathIA
Introduction: In this task I will consider a set of numbers that are presented in a symmetrical pattern and try to find a general equation to find the elements in the [pic]row.
Consider the five rows of number shown below.
Figure 1 Lacsap’s Fractions
The aim of this task is to find the numerator of the sixth row and to find the general statement for [pic]. Let [pic] be the [pic]element in the [pic]row, starting with r=0.
First, I will make a table of the numerator and the row number to show the relationship between the numerator and the row number.
Table 1 A table showing the relationship of rows and numerator
The difference between the numerator of row 1 and row 2 is 2, then the difference between the numerator of row 2 and row 3 is 3. The second difference for each row is 1 and it shows that this is a geometric sequence. So, I will start by finding the equation using the quadratic formula, [pic], in which x is the row number and y is the numerator.
First, I will plug in the numbers in the second row, which is 2 for x and 3 for y to try finding the quadratic equation of the Lacsap’s sequence. Then, I will plug in the numbers in the third row and it forms a simultaneous equation. I will use substitution method to solve the equation as there are two unknowns in the...

...• What were the most revolutionary social and economic developments of the last quarter of the nineteenth century?
• How did different groups of Americans respond to those changes and how effective were their responses?
• What role did government play in these developments?
In the late 1900s some of the most social and economic developments were railroads, steel oil, the type writer cash register, light bulb and agriculture. The development of the railroad made it easier for immigrant to come to this country for work. This meant that there were more group of different races and cultures in America. And in some states there became an over population and city workers like police and garbage men could not keep up with the demand of so many people. Some groups mover to open land for the Homestead Act. They had hope of farming and staying on the land for at least five years as agreed but the supply and demand of agriculture did sustain so many farmer moved off the land well before their five years. The government played many different role I deescalating some issues in American history. Women and children were being worked for long hours and getting paid a little bit of nothing in return for their hard work. So the government put labor laws into place that were to protect women and children. As oil, steel and railroad industries grew so did the levels of pollution. The government again put laws that were to protect animals and the earth so that there would not...

...
ANALYSIS
Physics has a lot of topics to cover. In the previous experiments, we discussed Forces, Kinematics, and Motions. In this experiment, the focus is all about Friction. Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction like fluid friction which describes the friction between layers of a viscous fluid that are moving relative to each other; dry friction which resists relative lateral motion of two solid surfaces in contact and is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces; lubricated friction which is a case of fluid friction where a fluid separates two solid surfaces; skin friction which is a component of drag, the force resisting the motion of a fluid across the surface of a body; internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation and sliding friction.
When surfaces in contact move relative to each other, the friction between the two surfaces converts kinetic energy into heat. This property can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to heat whenever motion with friction occurs, for example when a viscous fluid is stirred. Another important consequence of many types of friction can be wear,...