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Why Study Math? – The Mathematics of Finance – Interest – Part I

Admin — Fri, 30 Apr 2010 11:57:16 +0000

All right kids. So you hate math and you don’t care whether you do well in this subject or not. But know one thing. Mathematics is the language of money. That’s right. Whether we’re talking interest on CD’s or bonds, dividend yields on stocks, or returns on investment for a business venture, math is the lingua franca–or universal languge. So remember that if you don’t learn math well, you just might be giving an opportunity to some other hungry kid who wants to own more Jordan sneakers than you. Simply put: if you want the goodies in life, you best learn now that math can lead you to the land of fruit and nuts.

Indeed mathematics is the subject that explains how money accumulates and grows over time. What we’re talking about here is interest–compound interest specifically. You see, when you go to your neighborhood bank and deposit a sum of money, the bank pays you for your generosity in letting the bank use that money. What the bank pays you is called interest and the way this is calculated is with the compound interest formula. This formula is the portal, or gateway, to more elaborate financial calculations: annuities, perpetuities, mortgages, and other financial instruments all hinge on this formula. Because of its importance, the compound interest formula is a necessary part of every person’s know-how.

Let’s examine this formula using a basic example. Suppose you deposit $1,000 at your local bank. The bank is paying a healthy 6% interest for your funds. If the bank were to compound this money annually, then you would calculate your accumulated amount by using the formula A = P*(1 + i), where A = the accumulated amount, i = the interest rate or .06, and P = the principal or $1,000. When we plug these values into the formula we obtain A = $1,060. Thus at the end of the year you will have earned $60 in interest and your new balance will be $1,000 + $60 or $1,060.

If we keep this money with the bank for another year, we will receive interest on, not $1,000, but on $1,060. The accumulated amount at the end of the second year will be A = $1,060*(1 + .06) or $1,123.60. To find the interest received over the second year we need only subtract the balance at the beginning, or $1,060. Thus we obtain $63.60 as the interest received over year 2. Notice that this is $3.60 more than the interest received from the first year. This is where the term compound interest comes from. In essence we are compounding interest on top of interest to get more interest each year.

We could have obtained the balance at the end of the second year by simply using the formula
A = P*(1.06)^2. If we plug in $1,000 for P and do the calculation, we arrive at $1,123.60. If we want to know the balance at the end of the third year, we use
A = P*(1.06)^3. For the balance at the end of n years, we use A = P*(1.06)^n, and we thus arrive at the general compound interest formula.

In the next article, we will look at different compounding periods and the net effect on the accumulated value of your money. Yes it is wise to know math, particularly when it comes to the mathematics of finance. See you next time…

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By: Joe Pagano

Let’s Be Math Explorers

Admin — Wed, 28 Apr 2010 13:15:48 +0000

Everyone Loves Exploring – Whether your ideal place to explore is a deep dark jungle, a long, sandy deserted beach or a marketplace in a distant, exotic land, we all have an idyllic place that we could spend hours and hours exploring and making wonderful discoveries.

Children are certainly keen explorers too, perhaps even keener than adults. Children still have a natural curiosity and an unending desire to explore and discover. So let’s look at how that natural curiosity can be used to advance math skills.

We know that children love to explore and discover so how can we harness this to help them with math. The answer is simple – let them explore and discover in math. Give them numbers to play with, let them explore patterns, let them discover properties of numbers and shapes and measurements.

Best of all, when they learn this way they also develop a real understanding of these concepts because they haven’t just been told that this is the way it is, they have discovered it for themselves.

Math games are a great way to explore in math. When children play with numbers they learn about numbers. They discover patterns. They discover properties of numbers. They discover relationships between numbers. They discover number concepts.

Best of all, when they learn this way they also develop a real understanding of these concepts because they haven’t just been told that this is the way it is, they have discovered it for themselves.

Let’s do some exploring of our own now and look at a few games we can use to explore math concepts.

Higher or Lower is a great game for exploring number order and place value.

To begin select the upper and lower number limits, e.g. between 10 and 100.

One player selects a number and records it on a piece of paper.

The other player/s guesses what they think the number might be. For each guess the first player tells if the chosen number is higher or lower. Play continues until the number is guessed. A more challenging game would be to explore larger numbers or decimal numbers. This game is always a favorite no matter what age group I am teaching. I just adjust the size of the numbers to the ability of the students.

Grab and Group is a great way to explore division. Players take turns to grab a group of items (marbles, toothpicks, counters or even pens). They then attempt to make groups of 2’s with no remainders, then 3’s, 4’s, 5’s and 6’s. Players score points for each of the groups they can make, e.g. a player who grabs 9 can only make 3’s so they score 3 points. A player who grabs 12 would score 2+3+4+6=15 points as they could make groups of 2’s, 3’s, 4’s and 6’s. This game is also easily adjusted to the abilities of the students. Younger students will manipulate the items to see if the number is divisible by the given number. Older children will simply count the items then mentally calculate what the number is divisible by.

You can also have children explore properties of numbers by posing a question for them to answer.

- How many prime numbers are there between two given numbers, e.g. 1 and 100 or 200 and 300?
- How many times do you write the digit 7 when you write all of the numbers between two given numbers, e.g. 1 and 100 or 500 and 700?

For more games to get your kids or your students exploring math you could try a number version of Hangman using a complete computation.

Tic Tac Toe can also be played in number versions, e.g. use the numbers 1-9 to fill the grid. The first player to create a line that totals 15 is the winner.

There are many other math games that are ideal for exploring in math. Playing games in math offers so much more than just a bit of fun. These games also help to develop a deep understanding of math concepts and a positive attitude towards a sometimes unpopular subject. Don’t be surprised when you even hear, ‘Gee that was fun! Can we do it again?’

By: Teresa Evans

How to Learn Math

Admin — Tue, 27 Apr 2010 16:21:03 +0000

Learning math is quite different than learning other subjects, and it is certainly different than learning isolated procedural tasks, such as how to change a flat tire. Math needs to be learned step-by-step. If you do not completely understand a particular topic, and are unable to perform tasks involving that topic with ease, there is no point in moving on to a new topic that depends upon the first one. However, this is exactly what most schools do, although in many cases, they simply do not have a choice.

Whether you are learning math from a website, or from a book, or from a private tutor, it is important to not move past a particular topic until you are fully comfortable with the one you are working on. You will simply fall farther and farther behind, and will get more and more confused and frustrated.

Another important point to understand is that you need to work on material that is at your level. This can be humbling if you are very far behind in math, but there is no point in struggling to learn material that you are not yet ready for. Many high school students struggle to solve equations such as 7x – 9 = 16x + 13, when the real problem was that they never fully learned basic math like addition and subtraction.

Try to find material that is at a level that you completely understand, and begin studying math from that point forward. It is OK if it is many grades below level. If you study hard, you will be able to catch up very quickly, and more importantly, you will have a solid understanding of all the fundamentals.

It is important to study math every day if you want to get good at it. Do not just finish your homework, close your book, and say that you are done. You are not done. You need to think about the material, close your book and quiz yourself, and then think about it some more. Each day, you also have to go back and review earlier material.

Math is not about “doing,” it is about understanding. It is about thinking. Do not just answer a question. Ask yourself why the answer makes sense, and why the method that you used makes sense. If you study in this fashion, you will have no trouble at all passing your math tests, and you will probably get close to 100 on most of them. The math just has to become a part of you, and that can only happen if you put in a great deal of effort every single day. Think about how you learn a sport or a musical instrument. It takes tons of daily practice, and tons of concentration. Learning math is exactly the same.

By: Larry Zafran

Why Study Math? – The Hyperbola

Admin — Tue, 27 Apr 2010 13:23:30 +0000

As we continue the “Why Study Math” series of articles, here we look at the conic section called the hyperbola. The hyperbola is obtained by intersecting the double-napped cone (see the other articles in this series on this point) with a plane so that both parts of the cone are cut. Those familiar with the parabola might note that this curve almost looks like two parabolas pasted back to back with a space in between them. Mathematically, the hyperbola is not a parabola, although these two conic sections have a similar outward appearance.

The hyperbola is the least known of the four conic sections. It is also the most difficult curve to derive algebraically. Probably for this reason, students who study the conic sections, like the hyperbola the least. However, when students see the reason we study this curve, their attitude changes significantly. For this reason, we will now examine some of those applications connected to the hyperbola.

Everyone at one time or another has thrown a pebble into a still pond. Picture throwing not one but two pebbles into this pond at the same time. The outward concentric circles that form intersect each other at points which trace out the curve known as the hyperbola. This application is used in radar tracking stations. LORAN, the terrestrial navigation system, uses low frequency radio transmitters to locate objects. Objects are located by sending out sound signals from two sources to a receiving station, such as one found on a boat or plane. The constant time difference between the signals from the two stations is represented by a hyperbola.

As we discussed with the applications of the ellipse, most celestial bodies follow elliptical orbits. In the case of comets, however, a hyperbolic path is followed as they shoot through space. The hyperbola is also the shadow cast on a wall by a lamp with a cylindrical shade. And for something a little more earthy, the shape of that horse saddle you get on to ride forms an interesting solid curve called a hyperbolic paraboloid. So you see, the conic sections–even the hyperbola–might be closer than you think.

See more at Math Ebooks

By: Joe Pagano

SAT Writing, Math & Critical Reading – How to Significantly Improve Your Scores

Admin — Mon, 26 Apr 2010 13:23:31 +0000

The SAT can be a daunting and nerve-racking test for high school students all over California. After all, it’s a huge part of the journey on the way to applying to colleges! That’s why we’ve put together a list of ways you can significantly improve your score. Just follow these tips and you’re on your way! (Remember: You will have 3 hours and 45 minutes to take the SAT. Students with learning disabilities will receive time and a half.)

The Writing Section:

1. The highest score that you can receive on the SAT is 2400 (800 for the writing, critical reading, and math sections). The essay counts for 30% of your writing score. You will have 25 minutes to write a well-developed essay, for which you will receive a score of 1-6.

2. Decide if you agree or disagree with the essay prompt and write your thesis. Then give yourself no more than 5 minutes to quickly outline your essay response! Your essay should consist of 4-5 paragraphs: an introduction, one paragraph each detailing two or three different examples that support your thesis, and a conclusion. Try to use one example from literature / history / current events and your own personal experience.

3. After briefly outlining your essay, give yourself about 5 minutes to write each of the two or three supporting paragraphs and another 4 minutes to write your conclusion. Use the remaining time to proofread. This may seem overwhelming now, but the more you practice taking the essay portion of the SAT while timing yourself, the easier it will become!

4. Review the types of multiple-choice questions on the writing section: approximately 18 identifying sentence errors, 25 improving sentences, and 6 improving paragraph questions. Two sections will take 25 minutes each and 1 section will take 10 minutes. There is generally an unknown experimental section (an additional math, writing, or critical reading section) on the SAT that will not count toward your score.

5. Common errors tested are: subject-verb agreement, faulty parallelism, incorrect tenses, inappropriately used adjectives and adverbs, wordiness, irregular verbs, and idioms (common phrases/expressions). Examples of idioms are “better late than never” or “forget about it.” An answer choice may say “better late than forever” or “forget over it,” and you must determine that this answer is incorrect.

6. Since you will receive 1 point for correct answers and lose