Another person who contributed to the Pascal’s Triangle was Omar Khayyam. He was a great eleventh-century Indian astronomer, poet, philosopher, and mathematician, who lived in what is now Iran. He was born May 18^th in 1048 in Nishapur, Persia. Khayyam described the array of numbers in the future Pascal’s Triangle as a useful tool for representing the number of combinations of short and long sounds in poetic meters (see figure 2 for a glimpse of Pascal’s Triangle in Arabic numerals). He wrote several works, such as Problems of Arithmetic, a book on music and another on algebra. His studies of the future Pascal’s Triangle were the earliest records found of the triangle that would eventually bear Blaise Pascal’s name. He also contributed the Jalali calendar, new components to algebra, astronomical tables, and the Rubaiyat. He died December 4^th in 1131 at Nishapur, Persia.

In the seventeenth century, Blaise Pascal helped develop the theory of counting in the Pascal’s Triangle (see figure 3 for a picture of Pascal). He was a child prodigy who became interested in Euclid’s Elements at the age of twelve. Four years later, he was conducting original research and wrote a paper of such quality that some of the leading mathematicians of the time refused to believe that a sixteen-year-old boy was the author. His knowledge and wisdom about mathematics gave him the strength to create new theories and equations for the future.

Pascal created a theory, known as “Pascal’s Theorem,” which stated that if a hexagon were inscribed in a cone, the points of intersection of the opposite sides will lie in a straight line. He employed his arithmetic triangle in 1653 (see figure 4 for the original triangle), but no account of his method was printed until 1665. The triangle was constructed with each horizontal line being formed from the one above it by making every number in it equal to the sum of those above and to the left of it the row immediately above it.

Pascal also made other contributions to mathematics. He created the first digital calculator, known as the Pascaline to help his father, who was a tax collector. Adding French currency was difficult because the currency consisted of different coins, worth different values. Pascal’s machine, however, was not a great success. The only function it could perform was addition!

Pascal abandoned mathematics in his later years and devoted his time and life completely to philosophy and religion. In 1658, however, while being unable to sleep because of a toothache, he decided to think about geometry to take his mind off the pain and surprisingly, the pain stopped! Pascal took this as a sign from God and heaven that he should return to mathematics. But for a short time he returned to his research until he was seriously ill with dyspepsia, a digestive disorder. He lived the remaining years of his life in excruciating pain, doing little work until his death at age thirty-nine in 1662.

Pascal’s Triangle is an array of numbers that has numerous applications in math. It is not a geometric figure, but an array of natural numbers shaped in the form of a triangle. The sum of two adjacent numbers is equal to the number directly below and between them. The triangle continues infinitely. The numbers in horizontal lines make up rows. Numbers in an oblique line on the diagonal are called diagonals, or columns. They are numbered from the zero diagonal to infinity. Numbers in the Pascal’s Triangle are referred to as elements.

In order to construct Pascal’s Triangle, one must first know that entries in the triangle are given the row number and place within that row. Starting with row zero and place zero, the number one will always be at the top of the triangle and at the first and last entries in each row. In rows zero and one, there will always be ones in all the entries. For row three, you must employ the rule, which states that any number within the triangle is the sum of the two numbers immediately above it. Knowing that rule will help one construct the triangle perfectly.

For example, the triangle will start with the number one in row zero. Then, two ones are located in row one because the blank spots next to the number one in row zero are considered to be zeroes. In row two, the sequential order will be one, two and one because the number one must always be on the sides of the triangle and the number two is the sum of the numbers directly above it, which are one and one.

As on would imagine, with all these numbers in the triangle, there are many patterns present. There are a few that are obvious. One could be found by summing the numbers in each row. The sum of the numbers in any row equals 2ⁿ, where n is the number of the row. For example, in use row four, the sum of the numbers is sixteen and 2⁴ equals sixteen.

Another pattern involves the prime numbers. For any row whose second placed entry is prime, then all the numbers in that row, excluding the ones on the ends, are divisible by that prime number. For example, let’s use row seven. The numbers seven, twenty-one and thirty-five are all divisible by seven, the prime number. However, when the number is composite, the pattern will not work. In row eight, twenty-eight and seventy are not divisible by eight.

A third pattern within Pascal’s Triangle is known as the Hockey Stick Pattern (see figure 5 as an example). If a downward diagonal of numbers are selected, beginning with any of the ones bordering the sides of the triangle and ending as any number inside the triangle on that same diagonal, the sum of the numbers inside the selection is equal to the number below the last number in the selection that is not on the same diagonal. Using figure 5 as an example, one plus nine equals ten. One plus five plus fifteen equals twenty-one. One plus six plus twenty-one plus fifty-six equals eighty-four.

There are other types of sequences that can be found within Pascal’s Triangle. One such sequence is that the diagonals beginning from row one and extending downward (exclusive of the outer ones column) forms the counting numbers (ex. one, two, three, four, five…). Another sequence is demonstrated by the number of points required to make triangles of progressively greater sizes (ex. one, three, six, ten). That sequence can be found beginning with the “1” places within row two (one, three, six, ten, fifteen, twenty-one).

A special form of math that is patterned from Pascal’s Triangle is known as the Fibonacci sequence. This is applicable, in many natural mathematical situations ranging from sunflowers to pinecones. It is also the source of the Golden Ratio. The Golden Ratio is a special number approximated to 1.6180339887498948482. The Greek letter Phi represents it. Like Pi (approximates to 3.14), the digits of the Golden Ratio go on forever without repeating, also referred to as a nonrepeating decimal. The ratio of a term in the sequence to the term right before it approaches the Golden Ratio as the sequence reaches infinity.

Starting with one and adding the preceding numbers on the diagonal provide the next term for the Fibonacci sequence. For example, one plus zero equals one, one plus one equals two, one plus two equals three, two plus three equals five, and three plus five equals eight. In simpler form, the first six terms of the Fibonacci sequence, are one, one, two, three, five and eight. This sequence can be found in Pascal’s Triangle by taking the sums of the diagonals as shown in figure 6.

Pascal’s Triangle is used in many aspects of our life. One use is in probability, specifically within the field of medicine. Frequently doctors experiment with combinations of new drugs to combat hard-to-treat illnesses such as AIDS and hepatitis. Let’s assume that a drug company has developed five antibiotics and four immune-system stimulators to treat AIDS. Pascal’s Triangle can be applied to figure out how various treatment programs consisting of three antibiotics and two immune-system stimulators can treat the disease. Pascal’s Triangle can help speed our computations of various treatment protocols combinations.

To solve this problem, we must select the drugs in two ways: first the antibiotics and then the immune-system stimulators. With the antibiotics, we are choosing three drugs from five, which can be done in C (5,3) ways. Looking at the fifth row of Pascal’s Triangle, the third entry is ten, which makes C (5,3) equal ten ways. When choosing two immune-system stimulators from four, it can be done in C (4,2) ways. Looking at the fourth row of Pascal’s Triangle, the third entry is six, which makes C (4,2) equal six ways. Thus, to find the answer for the antibiotics and the immune-system stimulators together, the answer will be ten multiplied by six equals sixty ways.

Another application of Pascal’s Triangle involves formulation of combinations. Suppose that a person has five hats on a rack and they want to know how many different ways they can pick two of them and wear them. It does not matter to the person which hat is on top, it just matters which two hats are picked. We are trying to figure out how many ways that person can pick two objects from a set of five objects. The answer is the number in the second place in the fifth row of Pascal’s Triangle, the number ten. The binomial coefficient in this case is six to three or six choose three where the number ten is located. Pascal’s Triangle is like a grid, with ordered pairs, except the numbers are labeled with row, then entry.

Pascal’s Triangle is also used in algebra. Suppose that one has the polynomial of x plus one and they want to raise it a few powers, like one, two, three, four and five. If you make a chart of what you get when you do these power raisings (ex. (x + 1) ¹ = 1 + x), it will look like Pascal’s Triangle only with coefficients. Because of this connection, the entries in Pascal’s Triangle are called the binomial coefficients.

A binomial is an algebraic expression with two terms, (ex. x + y, 3x²z – 2siny) having either one addition or one subtraction sign. A coefficient is a number that is next to a variable (ex. In x + y, the coefficients are one and one and in the algebraic expression 3x²z – 2siny, the coefficients are 3 and –2). When inside the Pascal’s Triangle, the terms are multiplied by themselves to create binomial coefficients.

In conclusion, Pascal’s Triangle is an array of natural numbers that are used in algebra and probability, mostly to find combinations. The dedicated works of ancient scholars and mathematicians have brought math to a whole new dimension. Without their studies and hypotheses, our world would not be as modern as it is today.

Why this does usually come to students? Actually, knowing what the reasons for having low grades or failing math is not our main goal here but passing or becoming an ace.

Now, listed below are some tips about acing math:

1. Always listen carefully from the discussion of your mentor. Disliking your mentor's way of discussing or say his/her personality will surely result to failure.
2. Set a study schedule and follow it promptly and do not forget to study your other subjects, remember that math is not your only subject or course.
3. Answer problems from math books and assignments given by professors or teachers and set your mind to the problems not on anything else. Please be patient in solving problems.
4. Always attempt to answer any problem because if you just look at it and think that it is hard, you will just waste a lot of time. There is a saying in math that the answer is always in the attempt.
5. Seek for help, advice or wisdom from your professor, teacher or friend who is good in math when you feel that you cannot solve it. Do not be ashamed to ask them since this is for your own good.
6. Look for patterns while solving problems and relate it to the physical world because this will facilitate in remembering or storing it in your mind.
7. Study in advance or ahead from your teacher or professor's planned lesson to discuss.
8. Provide enough time for resting to release stress while at school or when solving problems. Play a basketball game, hangout with friends, etc.
9. Exercise daily to keep your body physically fit.
10. Finally, seek God's wisdom and knowledge by praying to Him and asking for guidance.

Those tips above are really effective but it won't work if you will not discipline yourself especially with the study schedule. And please be flexible and try to make your own way of becoming an ace in Math.

“Have a wonderful and exciting trip on becoming an ace in Math!”

For example: 6.454 is smaller than 6.7 because it has more digits. Researchers have found that there are two “rules” to help compare decimal places as well. One example the author give is that a child could just write out the units as thousands, hundreds, tens, ones, tenths, hundredths thousandth knowing that the decimal lies on the ones place. I am a very visual person and I feel that would be helpful when trying to figure out the place value of that number. The second way that was represented was by using base-ten-blocks. Using the blocks you can identify six-tenths of 100 by looking at the ten-by-ten-by-ten cube. This way is also very visual, I think all these ways are very helpful as long as the child understands the concept to the fullest of their ability before moving on to the next step.

1. Number e

   It is one of the most important numbers in Math, with Pi and imaginary number i. Number e is called “Number of Euler”, pertaining to Leonhard Euler, a mathematician who discovered it. This number, is also an irrational number. First decimal numbers of it are: 2.718281828459…
   
   Imagine that you have $1. And you deposit it into an imaginary box, that gives you 100% of your money the next year. One year later, you will have $2. But if you deposit it 6 months, you withdraw it and then deposit it for another six months, after a year you will have $2.25.
   
   The shorter you deposit it, the bigger the gain. The maximum you can gain is: e (2.7182…)

2. Number Pi

   This number is similar to e, irrational, so we cannot express it as a fraction. First decimal numbers of it are: 3.141592… Also, Pi is known as .
   
   Number Pi is probably the most known number, and one of the most important.
   
   This number has a lot of uses, like resolving the perimeter of a circle ( · d) and the area of a circle ( · r²).
   
   I.e. you have a circle of 2 meters of diameter and you want to resolve its perimeter. In this case, you have to do:
   
   ( · d) = 3.14 · 2.
   
   Per = 6.28.

3. Number Phi

   It is also called “the golden ratio” or “divine proportion”. It is an irrational number: 1.61803… and it is symbolized using the Greek letter Φ. Mathematicians, sculptors and intellectuals have studied this number for more than 2000 years.
   
   Number Phi appears in music, painting, architecture, nature, etc.
   
   You can find this number in the Fibonacci sequence, and in natural ratios.
   
   Phi can be found in:
   
   
   • Pyramids
   • Pentagram
   • Distribution of the leaves in the plants.
   • Relation between the length of our arm (fingers until shoulder) and the length until elbow.

Fielding:

Game One - Catcher (first 2 innings only)

Game Two - 3rd Base

Game Three - Left Field

Game Four - Shortstop (first 2 innings only)

Game Five - Left Center Field

Game Six - 2nd Base

Game Seven - catcher (last 2 innings only)

Game Eight - Right Center Field

Game Nine - Pitcher

Game Ten - Shortstop (last 2 innings only)

Game Eleven - Left Field

Game Twelve - 1st Base

Batting:

Start in batting position 1 and move down one position every game until you reach 12, then cycle back to 1 repeat.

Player Two

Fielding:

Game One - 3rd Base (first 2 innings only)

Game Two - Left Field

Game Three - Shortstop

Game Four - Left Center Field (first 2 innings only)

Game Five - Right Field

Game Six - Right Center Field

Game Seven - 3rd Base (last 2 innings only)

Game Eight - Pitcher

Game Nine -Right Field

Game Ten - Left Center Field (last 2 innings only)

Game Eleven - 1st Base

Game Twelve - Catcher

Batting:

Start in batting position 2 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 2, then repeat.

Player Three

Fielding:

Game One - Left Field

Game Two - Shortstop (first 2 innings only)

Game Three - Left Center Field

Game Four - 2nd Base

Game Five - Right Center Field (first 2 innings only)

Game Six - Pitcher

Game Seven - Right Field

Game Eight - Shortstop (last 2 innings only)

Game Nine - 1st Base

Game Ten - Catcher

Game Eleven - Right Center Field (last 2 innings only)

Game Twelve -3rd Base

Batting:

Start in batting position 3 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 3, then repeat.

Player Four

Fielding:

Game One - Shortstop

Game Two - Left Center Field (first 2 innings only)

Game Three - 2nd Base

Game Four - Right Center Field

Game Five - Pitcher (first 2 innings only)

Game Six -

Game Seven -

Game Eight -

Game Nine -

Game Ten -

Game Eleven -

Game Twelve -

Batting:

Start in batting position 4 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 4, then repeat.

Player Five

Fielding:

Game One -

Game Two -

Game Three -

Game Four -

Game Five -

Game Six -

Game Seven -

Game Eight -

Game Nine -

Game Ten -

Game Eleven -

Game Twelve -

Batting:

Start in batting position 5 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 5, then repeat.

Player Six

Fielding:

Game One -

Game Two -

Game Three -

Game Four -

Game Five -

Game Six -

Game Seven -

Game Eight -

Game Nine -

Game Ten -

Game Eleven -

Game Twelve -

Batting:

Start in batting position 6 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 6, then repeat.

Player 7

Fielding:

Game One -

Game Two -

Game Three -

Game Four -

Game Five -

Game Six -

Game Seven -

Game Eight -

Game Nine -

Game Ten -

Game Eleven -

Game Twelve -

Batting:

Start in batting position 7 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 7, then repeat.

Player Eight

Fielding:

Game One -

Game Two -

Game Three -

Game Four -

Game Five -

Game Six -

Game Seven -

Game Eight -

Game Nine -

Game Ten -

Game Eleven -

Game Twelve -

Batting:

Start in batting position 8 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 8, then repeat.

Player Nine

Fielding:

Game One -

Game Two -

Game Three -

Game Four -

Game Five -

Game Six -

Game Seven -

Game Eight -

Game Nine -

Game Ten -

Game Eleven -

Game Twelve -

Batting:

Start in batting position 9 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 9, then repeat.

Player Ten

Fielding:

Game One -

Game Two -

Game Three -

Game Four -

Game Five -

Game Six -

Game Seven -

Game Eight -

Game Nine -

Game Ten -

Game Eleven -

Game Twelve -

Batting:

Start in batting position 10 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 10, then repeat.

Player Eleven

Fielding:

Game One -

Game Two -

Game Three -

Game Four -

Game Five -

Game Six -

Game Seven -

Game Eight -

Game Nine -

Game Ten -

Game Eleven -

Game Twelve -

Batting:

Start in batting position 11 and move down one position every game until you reach 12, then cycle back to 1 and shift down again until you reach 11, then repeat.

Player Twelve

Fielding:

Game One - Catcher (last 2 innings only)

Game Two - 3rd Base

Game Three - Left Field

Game Four - Shortstop

Game Five - Left Center Field

Game Six - Second Base (first 2 innings only)

Game Seven - Second Base (last 2 innings only)

Game Eight - Right Center Field

Game Nine - Pitcher

Game Ten - Right Field

Game Eleven - 1st Base

Game Twelve - Catcher (last 2 innings only)

Batting:

Start in batting position 12, then go to one and rotate down one batting position at the start of every game until you reach twelve again.

The results regarding the study of the fifth and sixth grade students showed that after giving the students a written test consisting of three nonproblematic problems and six problematic (problems that can not be solved by a straightforward arithmetic operation) whose solution was ± 1 than the answer produced by adding or subtracting two numbers, the students answered 24% of the problematic items correctly and 83% of the errors on the problematic items were caused by ± 1. Of the strategies used to solve the problems, 78% of them were formal, 6% were informal, and 16% percent were unclear. The conclusion drawn here is the students whom used formal strategies show an understanding of the problem and necessary operation but those whom used counting based strategies show a better understanding of the mathematical situation but were unable to use the necessary operation. As for the teachers, the study called for them to complete a written test consisting on nine experimental items (3 solved by straightforward addition/subtraction, 6 problematic) and seven buffer items. Focusing on their strategies, solutions, and interpretations of solutions, teachers performed well on the nonproblematic items with 92% correct responses, however, they only scored 9% correct responses regarding the problematic items. Similar to the students, a large percent of the errors were due to ± 1 mistakes.

The results of these two studies can mean several things. Teachers and students may approach word problems in a superficial way because they are used to solving problems using a straightforward operation. Another explanation could be the lack of understanding with heuristic strategies or an enumeration process involving ordinal numbers. Overall, it seems both teachers and students need more practice in solving whole-number addition and subtraction word problems in which the solution is ± 1 than the sum/difference of the two numbers. The article suggests different strategies to improve on this weakness. The strategies consist of using a variety of instructional approaches, modeling problematic problems, presenting the problems first to students so in return they have the opportunity to develop their own strategies and providing students with mathematical tasks that require making connections between math and real-world knowledge.

After reading this article, I was surprised to read some of the results I did from the two studies regarding teachers and students. This article interested me before I even read it because of the topic of word problems. As a child and still to this day, I am far more capable of solving a mathematical operation by itself than one that needs to be drawn from a word problem. Often, it was just easier to take any two numbers within the problem and put them in the learned formula, but I never made the connection to why I was doing what I was doing. I enjoyed being able to read just how many people that affects and that students were not the only ones having difficulty, but the teachers too. I agree with the article when they stress the importance of showing students and teachers the relation math has to real-world knowledge and the reason they carry out certain operations. I believe students and teachers would be better off and have a better understanding of math and strong problem solving skills.

What's That in the Sky?

The A5's design is part sports car, part Jet Ski, and part airplane. The futuristic aircraft's folding wings tuck neatly under a slim rear tail so it can be towed behind a car. For an airplane, the A5 is small, about as long as two compact cars parked bumper to bumper, with a wingspan of 34 feet.

Cockpit Lite

Designers drastically simplified the cockpit experience most people know from images of commercial airplanes. A simple rack of gauges lines the cockpit's center console.

Good Old Gauges

Instead of supplying all the plane's gear on one digital screen-a common feature in private planes-the designers decided to stick with analog gauges to emphasize the plane's sense of reliability and to include a design motif familiar from autos.

“Amphibi-dextrous”

The A5 can take off from any standard runway and reach a height of 10,000 feet. The plane will also be available in an amphibious configuration, enabling it to take off from and land on the water.

My Plane Is in the Garage

ICON envisions making these light aircraft as common as Jet Skis. The plane can be washed with a hose and stored in a garage.

Flier-Friendly Rules

A new certification created by the FAA four years ago will make it easier for potential owners to learn to fly. The new Sport Pilot License requires just 20 hours of flight training and costs between $3,000 and $4,500-about half the time and cost of the previous easiest-to-obtain license.

Will It Fly?

The downturn in the economy has dampened discretionary spending, creating greater competition for consumers' leisure dollars. The A5 will be pitched as a luxury product at a time when consumers are cutting back. And steadily rising fuel prices aren't likely to help either. (The A5's engine, which burns either jet fuel or regular gasoline, gets about 18 miles to the gallon and has a range of about 300 miles.)

Another tip I learned while I was in Algebra 1 was how to find the square root of large numbers. My teacher always says look at what the number ends in. Even though this is not really a trick it can help you out a lot when trying to find a square root quickly. So let's say the number is 625 the number ends in a 5, so the only numbers that the square root can end in are only 5. Now you have to think a little bit for the tens digit. Say to yourself what number squared that ends in a 5 could be 625. 30 = 900 is too big, and 10 = 100 is too small. The number would be 25.

Last, a trick that will help you find how a large number is divisible by 11is to add up every other digit, (so you have two different sets of numbers) and subtract them. If the number you get after subtracting them equals 11 or 0, the number is divisible by 11. For example let's use 759. 7+9=16 while left over you have 5. 16-5=11 so 759 is divisible by 11. Now for bigger numbers like 7172 it is the same thing, you just have to add up more numbers.

This will help when you cannot find a number divisible by anything, and always try 11 because they can be trying to trick you.

Many are overwhelmed by the prospects of understanding math because of its terminologies and formulas such as with Algebra. But unknown to those same people who are confused about the matter, they use Algebra while functioning through their lives every day. Oh really…

Let us take for example, you have $1.00 and you want to buy a piece of candy at a store. When you purchase it, the cashier hands you back 50¢ in change. How much did the candy cost you?

Though you didn't consider it, this is a simple algebraic equation. Replacing the letter "c" for candy you can look at this as an equation:

$1.00 - c = $0.50

Replacing "c" for candy is what is called in math a "variable". Can you see how you may use Algebra and you didn't realize you were?

Let's try another example…

One of your co-workers gives you $8.00 and another co-worker gives you some money to buy them lunch, when you count the total money it is $17.00. How much did the second co-worker give you?

You probably figured this out without even the use of paper and pencil, yet this too Algebra. Simplifying the use of the word co-worker and substituting it with the variable "w", the equation is:

$8.00 + w = $17.00

Can you imagine how many algebraic equations you work out in your head whenever you buy groceries at a supermarket or shop at a mall. The reality is you don't think about these activities as equations;; in fact you typically don't put much thought into making these evaluations at all.

Please note the above examples are illustrations of simple algebra, but they are algebra nonetheless. Hopefully, you can see this is not as confusing as you may have first believed.

Of course I must advise, if you desire to continue the study of mathematics, the journey would eventually lead you to more intense math principles such as Calculus, which is used to help measure those distant planets, genetic codes, and other complicated equations.