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Submitted By charxiufan

Words 4670

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Words 4670

Pages 19

Introduction

Moral Values: I have learned many moral values while completing this assignment. Better still, I got to know about the importance of applying these moral values into our daily lives. The first moral value is to cooperate with other people. The strategies and solutions of the questions in this project work were discussed among me and a group of friends. This makes things easier and saved a lot of time. The management of time is also important to complete this project work. Other than this assignment, I have homework, extra co curricular activities and tuition classes to attend. Thus a good management of time is essential for me to complete this given task and not to disrupt my daily activities. Perseverance has taught me to be steady and persistent in doing something. In spite of many difficulties I faced throughout the whole procedure I learned that giving up is just not right solution. Obstacles and discouragement should be endured the course of action should be held on with unyielding determination to see obtain sweet fruit of success. I had also learned to appreciate the beauty of mathematics. Waxing eloquently on the basic importance of Mathematics in human life, Roger Bacon (1214-1294), an English Franciscan friar, philosopher, scientist and scholar of the 13th century, once stated: "Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world." And the ingenuity of his statement is there before us to see, in this Internet era.

Objectives:

My objectives to achieve upon completion of the Additional Mathematics Project Work are:

1. to realize the importance and the beauty of mathematics.

2. to prepare myself for the demands of my future undertakings and in workplace.

3. to use technology especially the ICT appropriately and effectively.

4. to acquire effective mathematical communication through writing, and to use the language of mathematics to express mathematical ideas correctly and precisely.

5. to apply and adapt a variety of problem-solving strategies to solve routine and non-routine problems.

Introduction to probabilities:

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such area of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and underlying mechanics of complex systems. The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later. history of probabilities:

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Etymology

Probable and likely and their cognates in other modern languages derive from medieval learned Latin probabilis and verisimilis, deriving from Cicero and generally applied to an opinion to mean plausible or generally approved.

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Origins

Ancient and medieval law of evidence developed a grading of degrees of proof, probabilities, presumptions and half-proof to deal with the uncertainties of evidence in court. In Renaissance times, betting was discussed in terms of odds such as "ten to one" and maritime insurance premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums.

The mathematical methods of probability arose in the correspondence of Pierre de Fermat and Blaise Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance. Christiaan Huygens (1657) gave a comprehensive treatment of the subject.

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18th Century

Jacob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's The Doctrine of Chances (1718) put probability on a sound mathematical footing, showing how to calculate a wide range of complex probabilities. Bernoulli proved a version of the fundamental law of large numbers, which states that in a large number of trials, the average of the outcomes is likely to be very close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is likely to be).

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Nineteenth century

The power of probabilistic methods in dealing with uncertainty was shown by Gauss's determination of the orbit of Ceres from a few observations. The theory of errors used the method of least squares to correct error-prone observations, especially in astronomy, based on the assumption of a normal distribution of errors to determine the most likely true value.

Towards the end of the nineteenth century, a major success of explanation in terms of probabilities was the Statistical mechanics of Ludwig Boltzmannand J. Willard Gibbs which explained properties of gases such as temperature in terms of the random motions of large numbers of particles.

The field of the history of probability itself was established by Isaac Todhunter's monumental History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865).

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Twentieth century

Probability and statistics became closely connected through the work on hypothesis testing of R. A. Fisher and Jerzy Neyman, which is now widely applied in biological and psychological experiments and in clinical trials of drugs. A hypothesis, for example that a drug is usually effective, gives rise to a probability distribution that would be observed if the hypothesis is true. If observations approximately agree with the hypothesis, it is confirmed, if not, the hypothesis is rejected.

The theory of stochastic processes broadened into such areas as Markov processes and Brownian motion, the random movement of tiny particles suspended in a fluid. That provided a model for the study of random fluctuations in stock markets, leading to the use of sophisticated probability models in mathematical finance, including such successes as the widely-used Black-Scholes formula for the valuation of options.

The twentieth century also saw long-running disputes on the interpretations of probability. In the mid-century frequentism was dominant, holding that probability means long-run relative frequency in a large number of trials. At the end of the century there was some revival of the Bayesian view, according to which the fundamental notion of probability is how well a proposition is supported by the evidence for it.

The mathematical treatment of probabilities, especially when there are infinitely many possible outcomes, was facilitated by Kolmogorov's axioms(1931). examples of the probability theory applications:

Weather forecasting

Suppose you want to go on a picnic this afternoon, and the weather report says that the chance of rain is 70%? Do you ever wonder where that 70% came from?

Forecasts like these can be calculated by the people who work for the National Weather Service when they look at all other days in their historical database that have the same weather characteristics (temperature, pressure, humidity, etc.) and determine that on 70% of similar days in the past, it rained.

As we've seen, to find basic probability we divide the number of favorable outcomes by the total number of possible outcomes in our sample space. If we're looking for the chance it will rain, this will be the number of days in our database that it rained divided by the total number of similar days in our database. If our meteorologist has data for 100 days with similar weather conditions (the sample space and therefore the denominator of our fraction), and on 70 of these days it rained (a favorable outcome), the probability of rain on the next similar day is 70/100 or 70%.

Since a 50% probability means that an event is as likely to occur as not, 70%, which is greater than 50%, means that it is more likely to rain than not. But what is the probability that it won't rain? Remember that because the favorable outcomes represent all the possible ways that an event can occur, the sum of the various probabilities must equal 1 or 100%, so 100% - 70% = 30%, and the probability that it won't rain is 30%.

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Batting averages

Let's say your favorite baseball player is batting 300. What does this mean?

A batting average involves calculating the probability of a player's getting a hit. The sample space is the total number of at-bats a player has had, not including walks. A hit is a favorable outcome. Thus if in 10 at-bats a player gets 3 hits, his or her batting average is 3/10 or 30%. For baseball stats we multiply all the percentages by 10, so a 30% probability translates to a 300 batting average.

This means that when a Major Leaguer with a batting average of 300 steps up to the plate, he has only a 30% chance of getting a hit - and since most batters hit below 300, you can see how hard it is to get a hit in the Major Leagues!

Part 1

Question:

The theory of probability has been applied in various fields such as market research, medical research, transportation, business, management and so on.

(a) Conduct research on the history of probability and give at least two examples on how the theory of probability is being applied in real life situations. Then, write an Introduction to this Project Work based on your findings. You may include the historical aspects, examples of the probability theory applications and its importance to real life situations.

(b) The probability theory can be divided into two categories: Theoretical Probabilities and Empirical Probabilities. Find out, discuss and write about the difference between the Theoretical and Empirical Probabilities. Problem statement:

1. What is the history of probability and how is the theory being applied into our daily lives?

2. What is the difference between the Theoretical and Empirical Probabilities? Strategy:

1. Search for information by surfing the Internet for the history of probability and how is the theory being applied into our daily lives and write and Introduction to this Project Work based on the findings.

2. The definition of Theoretical and Empirical Probabilities were looked at from the reference book in the school library.

3. The comparison of Theoretical and Empirical Probabilities were shown by using a solution to highlight the difference.

Solution:

B) Difference between the Theoretical and Empirical Probabilities

The term empirical means "based on observation or experiment." An empirical probability is generally, but not always, given with a number indicating the possible percent error (e.g. 80+/-3%). A theoretical probability, however, is one that is calculated based on theory, i.e., without running any experiments.

Empirical Probability of an event is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). It is based specifically on direct observations or experiences.

Theoretical Probability of an event is the number of ways that the event can occur, divided by the total number of outcomes. It is finding the probability of events that come from a sample space of known equally likely outcomes.

Comparing Empirical and Theoretical Probabilities: Karen and Jason roll two dice 50 times and record their results in the accompanying chart.

1.) What is their empirical probability of rolling a 7?

2.) What is the theoretical probability of rolling a 7?

3.) How do the empirical and theoretical probabilities compare? | Sum of the rolls of two dice | 3, 5, 5, 4, 6, 7, 7, 5, 9, 10,

12, 9, 6, 5, 7, 8, 7, 4, 11, 6,

8, 8, 10, 6, 7, 4, 4, 5, 7, 9,

9, 7, 8, 11, 6, 5, 4, 7, 7, 4,

3, 6, 7, 7, 7, 8, 6, 7, 8, 9 | | Solution:

1.) Empirical probability (experimental probability or observed probability) is 13/50 = 26%.

2.) Theoretical probability (based upon what is possible when working with two dice) = 6/36 = 1/6 = 16.7% (check out the table at the right of possible sums when rolling two dice).

3.) Karen and Jason rolled more 7's than would be expected theoretically. | | conclusion: Empirical and theoretical probabilities are both different ways to calculate the probability. The results of an experiment may manipulate the answer of empirical probability but theoretical probability will not be affected and may not even need to conduct an experiment.

Part 2

Question:

(a) Suppose you are playing the Monopoly game with two of your friends. To start the game, each player will have to toss the die once. The player who obtains the highest number will start the game. List all the possible outcomes when the die is tossed once. (b) Instead of one die, two dice can also be tossed simultaneously by each player. The player will move the token according to the sum of all dots on both turned-up faces. For example, if the two dice are tossed simultaneously and "2" appears on one die and "3" appears on the other, the outcome of the toss is (2, 3). Hence, the player shall move the token 5 spaces. Note: The events (2, 3) and (3, 2) should be treated as two different events. List all the possible outcomes when two dice are tossed simultaneously. Organize and present your list clearly. Consider the use of table, chart or even tree diagram.

Problem Statement:

1. What is the possible outcome when the die is tossed once?

2. What are the possible outcomes when 2 dices are tossed simultaneously?

Strategy:

1. The possible outcome when the die is tossed once each is figured out by knowing that all numbers on the die is possible to be the outcome.

2. The possible outcomes when 2 dices are tossed simultaneously is figured out by using a chart to list all the outcomes.

Solution:

a) {1, 2, 3, 4, 5, 6}

b) Chart Dice 2

6 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) 5 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) 4 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) 3 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) 2 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) 1 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) Dice 1 0 1 2 3 4 5 6

conclusion:

There are 6 possible outcomes for (a) which are {1, 2, 3, 4, 5, 6}. While there are 36 possible outcomes of (b) which are (1,6), (2,6), (3,6), (4,6), (5,6), (6,6), (1,5), (2,5), (3,5), (4,5), (5,5), (6,5), (1,4), (2,4), (3,4), (4,4), (5,4), (6,4), (1,3), (2,3), (3,3), (4,3), (5,3), (6,3), (1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (1,1), (2,1), (3,1), (4,1), (5,1) and (6,1).

Part 3

Question:

Table 1 shows the sum of all dots on both turned-up faces when two dice are tossed simultaneously.

(a) Complete Table 1 by listing all possible outcomes and their corresponding probabilities.

(b) Based on Table 1 that you have completed, list all the possible outcomes of the following events and hence find their corresponding probabilities:

A = {The two numbers are not the same)

B = {The product of the two numbers is greater than 36}

C = {Both numbers are prime or the difference between two numbers is odd) I) = {The sum of the two numbers are even and both numbers are prime)

problem statement:

1. What are the possible outcomes and their corresponding probabilities for the sum of all dots on both turned-up faces when two dice are tossed simultaneously?

2. What are the possible outcomes of the following events and their corresponding probabilities?

Strategy:

1. The possible outcomes are all listed in table form.

2. The probability is calculated by dividing the number of outcome with 36.

Solution:

a) Table 1 show the sum of all dots on both turned-up faces when two dice are tossed simultaneously. Sum of the dots on both turned-up faces (x) | Possible outcomes | Probability, P(x) | 2 | (1,1) | 1/36 | 3 | (1,2),(2,1) | 2/36 | 4 | (1,3),(2,2),(3,1) | 3/36 | 5 | (1,4),(2,3),(3,2),(4,1) | 4/36 | 6 | (1,5),(2,4),(3,3),(4,2),(5,1) | 5/36 | 7 | (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) | 6/36 | 8 | (2,6),(3,5),(4,4),(5,3),(6,2) | 5/36 | 9 | (3,6),(4,5),(5,4),(6,3) | 4/36 | 10 | (4,6),(5,5),(6,4) | 3/36 | 11 | (5,6),(6,5) | 2/36 | 12 | (6,6) | 1/36 |

b) Table of possible outcomes of the following events and their corresponding probabilities. Events | Possible outcomes | Probability,P(x) | A | {(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5) } | 3036 | B | ø | ø | C | P = Both number are primeP = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)}Q = Difference of 2 number is oddQ = { (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4),(3,6), (4,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), (6,5) }C = P U QC = {1,2), (1,4), (1,6), (2,1), (2,2), (2,3), (2,5), (3,2), (3,3), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,3), (6,5) } | 2236 | D | P = Both number are primeP = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)}R = The sum of 2 numbers are evenR = {(1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2(, (6,4), (6,6)}D = P ∩ RD = {(2,2), (3,3), (3,5), (5,3), (5,5)} | 536 | part 4 question: (a) Conduct an activity by tossing two dice simultaneously 50 times. Observe the sum of all dots on both turned-up faces. Complete the frequency table below.

Based on Table 2 that you have completed, determine the value of:

(i) mean;

(ii) variance; and

(iii) standard deviation of the data.

(b) Predict the value of the mean if the number of tosses is increased to 100 times.

(c) Test your prediction in (b) by continuing Activity 3(a) until the total number of tosses is 100 times. Then, determine the value of:

(i) mean;

(ii) variance; and

(iii) standard deviation of the new data.

Was your prediction proven? problem statement:

1. What is the sum of all dots on both turned-up faces when two dice were tossed simultaneously for 50 times?

2. How to predict the value of the mean if the number of tosses is increased to 100 times?

3. How to prove the prediction?

Strategy:

1. A table is drawn to list down the sum of the two numbers (x), frequency (f), fx and fx2.

2. The mean, variance and standard deviation were calculated using mathematical skills learned in school.

3. Another table is drawn to list down the sum of the two numbers (x), frequency (f), fx and fx2 for 1000 tosses.

3. The mean, variance and standard deviation were calculated for the second table to prove the prediction. solution: a) Sum of the two numbers (x) | Frequency (f) | fx | fx2 | 2 | 2 | 4 | 8 | 3 | 4 | 12 | 36 | 4 | 4 | 16 | 64 | 5 | 9 | 45 | 225 | 6 | 4 | 24 | 144 | 7 | 11 | 77 | 539 | 8 | 4 | 32 | 256 | 9 | 6 | 54 | 486 | 10 | 3 | 30 | 300 | 11 | 1 | 11 | 121 | 12 | 2 | 24 | 288 | f = 50 | fx = 329 | fx2= 2467 |

Table 2

i) Mean = x = fxf = 32950 = 6.58 ii) Variance = = fx2f - x 2 = 246750 – (6.58)2 = 6.044 iii) Standard deviation = ( fx2f - x 2 ) = 6.0436

= 2.458

b)

Sum of the two numbers (x) | Frequency (f) | fx | fx2 | 2 | 4 | 8 | 16 | 3 | 5 | 15 | 45 | 4 | 6 | 24 | 96 | 5 | 16 | 80 | 400 | 6 | 12 | 72 | 432 | 7 | 21 | 147 | 1029 | 8 | 10 | 80 | 640 | 9 | 8 | 72 | 648 | 10 | 9 | 90 | 900 | 11 | 5 | 55 | 605 | 12 | 4 | 48 | 576 | f = 100 | fx = 691 | fx2= 5387 |

Prediction of mean = 6.91 i. Mean = 691100 = 6.91 ii. Variance = fx2f - x 2 = 5387100- (6.91)2 = 6.122 iii. Standard deviation = 6.122 = 2.474 Prediction is proven.

Part 5 question: When two dice are tossed simultaneously, the actual mean and variance of the sum of all dots on the turned-up faces can be determined by using the formulae below:

(a) Based on Table 1, determine the actual mean, the variance and the standard deviation of the sum of all dots on the turned-up faces by using the formulae given.

(b) Compare the mean, variance and standard deviation obtained in Part 4 and Part 5. What can you say about the values? Explain in your own words your interpretation and your understanding of the values that you have obtained and relate your answers to the Theoretical and Empirical Probabilities.

(c) If n is the number of times two dice are tossed simultaneously, what is the range of mean of the sum of all dots on the turned-up faces as n changes? Make your conjecture and support your conjecture. problem statement:

1. What is the actual mean, the variance and the standard deviation of the sum of all dots on the turned-up faces?

2. What are the comparisons between the mean, variance and standard deviation obtained in Part 4 and Part 5? strategy: 1. The actual mean, the variance and the standard deviation of the sum of all dots on the turned-up faces using mathematical methods.

2. The mean, variance and standard deviation obtained in Part 4 and Part 5 were tabulated and compared. solution: a)

Mean = x P(x)

= 2136+3118+4112+519+6536+716+8536+919+10112+11118+ 12136

= 7

Variance = xP(x) – (mean)

= 22136+32118+42112+5219+62536+7216+82536+9219+102112+ 112118+122136 - (7)2

= 54.83 – 49

= 5.83

Standard deviation = 5.83 = 2.415

b) | Part 4 | Part 5 | | n = 50 | n = 100 | | Mean | 6.58 | 6.91 | 7.00 | Variance | 6.044 | 6.122 | 5.83 | Standard deviation | 2.458 | 2.474 | 2.415 |

We can see that, the mean, variance and standard deviation that we obtained through experiment in part 4 are different but close to the theoretical value in part 5.

For mean, when the number of trial increased from n=50 to n=100, its value get closer (from 6.58 to 6.91) to the theoretical value. This is in accordance to the Law of Large Number. We will discuss Law of Large Number in next section.

Nevertheless, the empirical variance and empirical standard deviation that we obtained i part 4 get further from the theoretical value in part 5. This violates the Law of Large Number. This is probably due to a. The sample (n=100) is not large enough to see the change of value of mean, variance and standard deviation. b. Law of Large Number is not an absolute law. Violation of this law is still possible though the probability is relative low. conclusion: The empirical mean, variance and standard deviation can be different from the theoretical value. When the number of trial (number of sample) getting bigger, the empirical value should get closer to the theoretical value. However, violation of this rule is still possible, especially when the number of trial (or sample) is not large enough.

c)

The range of the mean

Conjecture: As the number of toss, n, increases, the mean will get closer to 7. 7 is the theoretical mean.

Image below support this conjecture where we can see that, after 500 toss, the theoretical mean become very close to the theoretical mean, which is 3.5. (Take note that this is experiment of tossing 1 die, but not 2 dice as what we do in our experiment)

FURTHER EXPLORATION In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

For example, a single roll of a six-sided die produces one of the numbers 1, 2, 3, 4, 5, 6, each with equal probability. Therefore, the expected value of a single die roll is

According to the law of large numbers, if a large number of dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the accuracy increasing as more dice are rolled. Similarly, when a fair coin is flipped once, the expected value of the number of heads is equal to one half. Therefore, according to the law of large numbers, the proportion of heads in a large number of coin flips should be roughly one half. In particular, the proportion of heads after n flips will almost surely converge to one half as approaches infinity. Though the proportion of heads (and tails) approaches half, almost surely the absolute (nominal) difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as number of flips becomes large. Also, almost surely the ratio of the absolute difference to number of flips will approach zero. Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows. The LLN is important because it "guarantees" stable long-term results for random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will converge to the expected value or that a streak of one value will immediately be "balanced" by the others.

In conclusion, the law of large numbers is more frequently used in casinos or calculations than empirical and theoretical probabilities due to its long-term stability and accuracy.

REFLECTION

While I was conducting the project, I learned the moral values that I practiced to complete this Project Work. This Project Work had taught me to be more confident when doing something especially the homework given by the teacher. I also learned to be a disciplined type of student which is always sharp on time while doing some work, complete the work by myself and researching the information from the internet.

With the information obtained while completing this assignment, I get to know the usefulness and importance of the possibility theory in calculations. Many ways to calculate probability were also learned via research using the Internet.

The End!…...

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... |Concept: Regrouping during multiplication of two-digit numbers without | | |decimals | |Subject: Mathematics |Grade: Fourth Grade, Whole Class | |State Standard [Virginia SOL]: | |VA-SOL 4.4 b > The student will... | |b) add, subtract, and multiply whole numbers; | |d) solve single-step and multistage addition, subtraction, and multiplication problems with whole numbers. | |National Standard | |Grades 3–5 Expectations: | |In grades 3-5 all students should- ...

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...Week One Videos Nongraded Activities and Preparation PhoenixConnect Learning Team Instructions Learning Team Charter Individual ® MyMathLab Exercises Individual Week One Study Plan Read Ch. 5, sections 5.2–5.4 and 5.6 of Beginning and Intermediate Algebra With Applications and Visualization. Read the University of Phoenix Material: MyMathLab Study Plan. Participate in class discussion. Respond to weekly discussion questions. Resource: University of Phoenix Material: Using MyMathLab ® Log on to MyMathLab on the student website. ® Complete the MyMathLab Orientation exercise. ® ® 4/15/13 4/15/13 4/15/13 2 2 Watch this week’s videos located on your student website. Follow the Math Help Community in PhoenixConnect. The focus of the community is to help students succeed in their math courses. Post questions and receive answers from other students, faculty, and staff from the Center for Mathematics Excellence. Resource: Learning Team Toolkit Complete the Learning Team Charter. Complete the Week One assignment in MyMathLab . ® 4/15/13 6 Review your Study Plan in MyMathLab after completing the homework assignment for the week. Select each topic from Ch. 5 in your study plan that has been highlighted with a pushpin • for further review. ® 4/15/13 1 First, complete some Practice problems until you feel ready for a Syllabus 3 MTH/209 Version 6 quiz. o Click the green Practice button within Objectives to Practice and Master. o Complete......

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...Sara Feliciano I have chosen to write my topic paper on ADD/ADHD. There is a lot of controversy surrounding this topic, the treatments that are used as well as the side effects. I found several of the other topics interesting as well but I have had personal experience with ADD/ADHD and think it would be beneficial for myself to learn more on this topic. When I was younger I was diagnosed with ADD. I was put on a daily dose of Adderall. I had reactions to this medication that I considered very bad, though could have been a lot worse. The side effect that most concerned me was that my heart rate had sped up significantly and I was constantly scared that something would seriously happen to me. I also had a minor side effect where I would constantly chew and had to have gum or I was chewing the insides of my mouth. I only took the medication for a short time because of how badly it bothered me but later I found out that there are different types of medications and I was put on the wrong one. I should have been put on the non-stimulant form rather than the stimulant. My brother was also diagnosed with ADHD when he was younger. He was also put on medications though I don’t think he was put on the same one as I. He had a different reaction, one that if I understand correctly is quite common. He completely changed while he was on it. He would do nothing. He would just sit and stare. He did have an easier time completing and focusing on schoolwork but he was not......

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...MATH 55 SOLUTION SET—SOLUTION SET #5 Note. Any typos or errors in this solution set should be reported to the GSI at isammis@math.berkeley.edu 4.1.8. How many diﬀerent three-letter initials with none of the letters repeated can people have. Solution. One has 26 choices for the ﬁrst initial, 25 for the second, and 24 for the third, for a total of (26)(25)(24) possible initials. 4.1.18. How many positive integers less than 1000 (a) are divisible by 7? (b) are divisible by 7 but not by 11? (c) are divisible by both 7 and 11? (d) are divisible by either 7 or 11? (e) are divisible by exactly one of 7 or 11? (f ) are divisible by neither 7 nor 11? (g) have distinct digits? (h) have distinct digits and are even? Solution. (a) Every 7th number is divisible by 7. Since 1000 = (7)(142) + 6, there are 142 multiples of seven less than 1000. (b) Every 77th number is divisible by 77. Since 1000 = (77)(12) + 76, there are 12 multiples of 77 less than 1000. We don’t want to count these, so there are 142 − 12 = 130 multiples of 7 but not 11 less than 1000. (c) We just ﬁgured this out to get (b)—there are 12. (d) Since 1000 = (11)(90) + 10, there are 90 multiples of 11 less than 1000. Now, if we add the 142 multiples of 7 to this, we get 232, but in doing this we’ve counted each multiple of 77 twice. We can correct for this by subtracting oﬀ the 12 items that we’ve counted twice. Thus, there are 232-12=220 positive integers less than 1000 divisible by 7 or 11. (e) If we want to exclude the......

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...TABLE OF CONTENT TITLE | PAGE | TABLE OF CONTENT | 1 | APPRECIATION | 2 | OBJECTIVE | 3 | METHOD INVESTIGATION | 4 | PART I | 5-7 | PART II | 8-10 | PART III | 11-13 | PART IV | 14-16 | REFLECTION | 17-18 | CONCLUSION | 19 | APPRECIATION First of all, I would like to say thank you for giving me the power to complete this project work. Not forgotten my parents for providing me any material to complete this project and their support which are the most needed for this project. Internet, books, computer and the list goes on. They also supported me and encouraged me to complete this project so that I will not procrastinatein doing it. For their strong support, I would like to express my gratitude to my beloved parents. They also helped me to find the mark to complete this project. They have always been at my side and I hope that they will still be there in the future. Besides, I would like to thank my Additional Mathematics teacher, Pn. Phong Bee Bee as she had gave us some important guidance and commitment during this project work. She has been a very supportive figure throughout the project. Then I would also want to thank my teacher for guiding me and my friends throughout this project. We had faced some difficulties in doing this task, but she taught us patiently until we knew what to do. She had tried to teach us until we understand what we supposed to do with the project work. I also want to thank my friends for assisting me to......

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...ADD is situational: in the same individual its expression may vary greatly from one circumstance to another. There are certain classes, for example, in which the ADD child may perform remarkably well, while in others she is scattered, unproductive, and perhaps disruptive. Teachers may conclude that the child is wilfully deciding when, or when not, to buckle down and work diligently. Many children with ADD are subjected to overt disapproval and public shaming in the classroom for behaviours they do not consciously choose. These children are not purposively inattentive or disobedient. There are emotional and neurophysiological forces at play that do the actual deciding for them. We shall examine them in due course. The second nearly ubiquitous characteristic of ADD is impulsiveness of word or deed, with poorly-controlled emotional reactivity. The adult or child with ADD can barely restrain himself from interrupting others, finds it a torture awaiting his turn in all manner of activities, and will often act or speak impulsively as if aforethought had never been invented. The consequences are predictably negative. One is forever trying to shut the barn door after the horse has bolted. "I want to control myself," a 33-year old man said at his first visit to my office, "but my mind won’t let me." The impulsiveness may express itself as impulse buying, the purchase of unneeded items on a sudden whim without regard for cost or consequence. "Impulse buying?", another man exclaimed......

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...Compilation of Different Math Verbal Problems Number Problems: 1. Consecutive. The sum of two consecutive integers is 15. Find the numbers. Solution: I will represent the first number by "n". Then the second number has to be "n + 1". Their sum is then: n + (n + 1) = 15 2n + 1 = 15 2n = 14 n = The exercise did not ask me for the value of the variable n; it asked for the identity of two numbers. So my answer is not "n = 7"; the actual answer is: "The numbers are 7 and 8." 2. Consecutive Odd. The lengths of the sides of a triangle are consecutive odd numbers. What is the length of the longest side if the perimeter is 45? Solution: Let x = length of shortest side x + 2 = length of medium side x + 4 = length of longest side Plug in the values from the question and from a sketch. 45 = x + x + 2 + x + 4 Combine like terms 45 = 3x + 6 Isolate variable x 3x = 45 – 6 3x = 39 x =13 Check your answer 13 + 13 + 2 + 13 + 4 = 45 3. Consecutive Even. The product of two consecutive negative even integers is 24. Find the numbers. Solution: (n)(n + 2) = 24 n2 + 2n = 24 n2 + 2n – 24 = 0 (n + 6)(n – 4) = 0 Then the solutions are n = –6 and n = 4. Since the numbers I am looking for are negative, I can ignore the "4" and take n = –6. Then the next number is n + 2 = –4, and the answer is The numbers are –6 and –4. ...

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...Name: Sharon Mohammed Candidate Number: School: Iere High School Class: Form 5s (Science) Project Title: Projectile Motion Additional Mathematics SBA: Teacher: Ms. Lia Chunilal Date: TABLE OF CONTENTS Project Title……………………………………………………………………………pg3 Aim of Project………………………………………………………………………..pg4 Problem Statement……………………………………………………………………pg5 Information about Volleyball…………………………………………………………pg9 Apparatus and Materials………………………………………………………………pg6 Method for Experiment……………………………………………………………….pg7 Solution to Problem…………………………………………………………………..pg8 Verification of Solution……………………………………………………………….pg18 Discussion……………………………………………………………………………..pg19 Conclusion…………………………………………………………………………….pg21 Bibliography...................................................................................................................pg22 PROJECT TITLE: To determine the ideal angle at which Penelope, a national volleyball player, needs to serve the ball in order to obtain the distance that it needs to go over the net but stay in the court y applying the theory of projectile motion. AIM: To determine the ideal angle and distance needed for a volleyball ball to be served over the net within the dimension of the court using projectile motion. PROBLEM STATEMENT: Penelope is a national volleyball player and she wants to know the angle at which she needs to serve the volleyball in order to get the ball over the net but also in the court every time she......

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...additional mathematics internal assessment [pic][pic] Title: To determine if the reaction time of an athlete is greater than the reaction time of a non athlete. Name of Candidate: Shamoy Cole Center #: 100033 Candidate #: 1000331331 Territory: Jamaica Teacher: Mr. Jarrett Acknowledgement I must first thank God the Lord God for blessing me with the knowledge, wisdom, strength and understanding in completing my Internal Assessment. I would also like to thank my teacher Mr. Leemore Jarrett who has been committed in his efforts to competently prepare me for this Mathematics examination. Last but not least, I would like to thank all my volunteers for their kindness in donating the time and require data that were fundamental in completing this assignment. Thanks to you all!!! Introduction This is a simple experiment to determine if the reaction time of an athlete is greater than the reaction time of a non athlete. This test uses the known properties of gravity to determine how long it takes a person to respond to the dropping of an object by measuring how far the object can fall before being caught. In this experiment the stimulus is visual (the dropping of a ruler) and the evidence of reaction is the catching of the ruler. The conclusion to be drawn will be done through careful analysis of simple statistical values to be completed from data collected. Research......

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...http://sahatmozac.blogspot.com ADDITIONAL MATHEMATICS MODULE 1 FUNCTIONS Organized by Jabatan Pelajaran Pulau Pinang 2006 http://mathsmozac.blogspot.com http://sahatmozac.blogspot.com CHAPTER 1 : FUNCTIONS Contents 1.1 Concept map Page 2 1.2 Determine domain , codomain , object, image and range of relation 3 1.3 Classifying the types of relations 3 2.1 Recognize functions as a special relation. 2.2 Expressing functions using function notation. 2.3 Determine domain , object , image and range 4-5 3.0 Composite Functions 6 -9 4.0 SPM Questions 9 – 10 5.0 Assessment test 11 – 12 6.0 Answers 13 – 14 http://mathsmozac.blogspot.com 1 http://sahatmozac.blogspot.com CONCEPT MAP FUNCTIONS Relations Object images ………….,, …………… …………… Functions Function Notation Type of relation y Or ……………… Composite Functions Inverse Functions f: x One to one Many to one ……….. fg ( x ) = ……………. Object f(x)=y ……………… http://mathsmozac.blogspot.com 2 http://sahatmozac.blogspot.com 1.1 Functions Express the relation between the following pairs of sets in the form of arrow diagram, ordered pair and graph. Arrow diagram Ordered pair Graph a ) Set A = Kelantan, Perak , Selangor Set B = Shah Alam , Kota Bharu ,Ipoh Relation: ‘ City of the state in Malaysia ‘ b )Set A = triangle,rectangle, pentagon Set B = 3,4,5 Relation : ‘......

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...The Difference between ADD and ADHD http://www.understood.org Ra The website (www.understood.org.) is intended for informational and instructive purposes. Accuracy can be viewed by its Copyright © 2014-2015. It does not constitute medicinal or lawful guidance. Their objective is to help families with minor’s ages 3and up to 20 years old, which are battling with learning disabilities and significance issues. The website authority is the most essential criteria to search for when discovering tenable material. It is critical for a creator of the webpage to be trustworthy, particularly when giving material on a sensitive subject. A professional in her field Rayma Griffin, has dedicated forty years of her vocation to operating with youngsters with learning disabilities within the schoolroom and as an administrator within the public and personal subdivisions (Rayma Griffin, on Jun 20, 2014). Rayma Griffin earned a double degree in psychology and specialized curriculum from the University of Massachusetts Amherst. Rayma Griffin holds a graduate degree in special education and a 6th year degree in organizational leadership from the University of Connecticut. At present, Rayma Griffin is an instructive specialist advocating for the rights of children who have learning inabilities and ADHD. On Jun 20, 2014 posted by Rayma Griffin, She explains the distinction between ADD and ADHD. ADHD stands for attention deficit hyperactivity disorder and ADD stands for attention......

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