## Making Mathematics Meaningful to Teenagers

It is important for students to find meaning in what they are learningÂ because meaning is one of the criteria the brain uses to identify information for long-term storage.Â One way to help learners find meaning is to connect what they are learning to their daily life. YetÂ too often students in secondary school mathematics classes have difficulty seeing the practical andÂ concrete applications of mathematics to Â everyday living. Here are just a few suggestions as to howÂ mathematical concepts can be meaningfully related to common experiences.

## Probability

**Determining odds.** Millions of people visit casinos, buy lottery tickets, play the stockÂ market, join in the office football pool, and meet with friends for a game of poker. They invest their money in chance, believing they can beat the odds. The mathematical principle of probability can tell us how often we are likely to win, helping us decide whether to riskÂ the odds and our money.

How do we determine probability? Letâ€™s say there are 12 apples in a fruit basket. FiveÂ are red and seven are green. If you close your eyes, reach into the basket, and grab one apple,Â what is the probability that it would be a red apple? Five of the 12 apples are red, so yourÂ chances of picking a red apple are 5 out of 12, or as a fraction, 5/12, which is about 42Â percent. Or, if you are choosing between two colleges, one in Texas and one in Connecticut.Â You decide to flip a coin. The chances are one out of two, or 1/2, of getting heads or tails.Â The odds are 50 percent for each.

What could the odds be for winning the state lottery if you buy only one ticket?

**Does gambling pay off? Odds in roulette.** Is roulette a good bet at a casino? Actually, theÂ casino will win more often than the player. Hereâ€™s why. The roulette wheel is divided into 38Â numbered slots. Two of these slots are green, 18 are red, and 18 are black. To begin the round,Â the wheel is spun, and a ball is dropped onto its outside edge. When the wheel stops, the ballÂ drops into 1 of the 38 slots. Players bet on which slot they believe the ball will land in. If youÂ bet your money that the ball will land in any of the 18 red slots, your chances of winning areÂ 18 out of 38, or about 47 percent. If you bet your money on a certain number, such as the redÂ slot numbered with a 10, your chances of winning fall to 1 in 38, or 2.6 percent.

The mathematics of probability guarantee that the roulette wheel will make money evenÂ if the casinoÂ doesn’tÂ win every time. Remember there are 18 each of the red and black slots.Â There are also 2 green slots. Whenever the ball lands in one of those green slots, the house Â wins everything that was bet on that round. So again, letâ€™s say you bet that the ball will landÂ in a red or black slot. This is the safest possible bet in roulette, since the odds are 18 out ofÂ 38 (47 percent) that you will win. But there are 20 out of 38 chances (53 percent) that youÂ will lose.

## Calculating Interest on Buying a Car

**How much are you actually paying when you finance a car purchase?** UnderstandingÂ interest can help you manage your money and Â help you determine how much it will cost youÂ to borrow money to pay for your car purchase. Interest is expressed as a rate, such as threeÂ percent or 18 percent. The dollar amount of the interest you pay on a loan is figured byÂ multiplying the money you borrow (called the principal) by the rate of interest.

Suppose you want to buy a used car for $10,000. The car salesman says that theÂ dealership will finance your car at a rate of 8.4 percent, and estimates your monthlyÂ payments at about $200 over a period of five years. How much money are you actuallyÂ paying back to the dealer over the term of the loan? Is this a good deal, or should you shopÂ around? What if a bank offered to loan you the $10,000 at a rate of 9.0 percent for fourÂ years? Which offer is better?

## Exponential Changes/Progressions

** Population growth.** The number of people living on Earth has grown dramatically in theÂ last few centuries. There are now 10 times more people on our planet than there were 300Â years ago. How can population grow so fast? Think of a family tree. At the top are twoÂ parents, and beneath them the children they had. Listed beneath those children are theÂ children they had, and so on down through many generations. As long as the familyÂ continues to reproduce, the tree increases in size, getting larger with each passing generation.

This same idea applies to the worldâ€™s population.

New members of the population eventually produce other new members so that theÂ population increases exponentially as time passes. Population increases cannot continueÂ forever. Living creatures are constrained by the availability of food, water, land, and otherÂ vital resources. Once those resources are depleted, a population growth will plateau, or evenÂ decline, as a result of disease or malnutrition.

How fast will population grow? Arriving at a reasonable estimate of how the worldâ€™sÂ population will grow in the next 50 years requires a look at the rates at which people areÂ being born and dying in any given period. If birth and death rates stayed the same across theÂ years in all parts of the world, population growth could be determined with a fairly simple Â formula. But birth and death rates are not constant across countries and through timeÂ because disease or disaster can cause death rates to increase for a certain period. A boomingÂ economy might mean higher birth rates for a given period.

The rate of Earthâ€™s population growth is slowing down. Throughout the 1960s, theÂ worldâ€™s population was growing at a rate of about two percent per year. By 1990, that rateÂ was down to one and a half percent, and is estimated to drop to one percent by the year 2015.Â Family planning initiatives, an aging population, and the effects of diseases such as AIDSÂ are some of the factors behind this rate decrease. Even at these very low rates of populationÂ growth, the numbers are staggering. Can you estimate how many people will be living onÂ Earth in 2015? By 2050? Can the planet support this population? When will we reach theÂ limit of our resources? How could this affect the lifestyle of your children or grandchildren?

** Is this job offer a good deal?** Looking to make a million dollars? Let us examine a plan forÂ earning a million dollars based on a contract between an employee and an employer. FirstÂ let us agree upon a contract.

Contract for Employment

Employee ____________ (enter your name)

Employer ____________ (a company agreeing with these terms)

Points of Agreement

1. The employee will work a five-day work week.

2. The employee will be paid for the weekâ€™s wages each Friday.

3. The employee will be hired for a minimum of 30 work days.

4. The salary schedule is as follows:

- The base pay for Day 1 is one penny.
- Â Each subsequent day, the salary is double that of the previous day.

Signed ____________________________ (Employee)

Signed ____________________________ (Employer)

Date: _______________________

Is this a good deal? Take a guess how much money this employee will have earned inÂ the 30 working days: My guess: $__________________. Calculate the amount one would earnÂ working six weeks (40 hours a week) at minimum wage? Minimum wage salary (beforeÂ taxes and other deductions) $___________. Now letâ€™s calculate the earnings for this contract andÂ see whether the employer or the employee has made the better deal. In week one, the wagesÂ would be: Monday, 1 cent; Tuesday, 2 cents; Wednesday, 4 cents; Thursday, 8 cents; and Friday,Â 16 cents, for a total weekly earnings of 31 cents. Doesnâ€™t seem like much does it? Now continueÂ calculating the daily wages for the next five weeks.

There is a formula that allows one to calculate a particular dayâ€™s wages without having to calculateÂ every step. This is an example of a geometric progression, a sequence of numbers in whichÂ the ratio of any number to the number before it is a constant amount, called the common ratio. ForÂ example, the sequence of numbers 1, 2, 4, 8, 16, … has a common ratio of 2. A geometric progressionÂ may be described by calling the first term in the progression X (in this example X is one cent),Â the common ratio as R (in this example, R = 2), and in a finite progression, the number of termsÂ as n. Then the nth term of a geometric progression is given by the expression: Xn = X1Rnâ€“1

Questions about this job:

1. How does the total amount of money earned compare with your original guess?

2. Suppose you wanted to buy a car. On which day could you purchase your car and pay inÂ cash?

3. Can you develop a formula for the daily salary? (Answer: Daily Salary = 2nâ€“1 X where n =Â the number of days youâ€™ve been working, and X = your base pay on Day 1.

This counting principle can be applied also to social causes. Efforts to address social issues areÂ often started by just a scant few individuals who are committed to a cause. Suppose you tell oneÂ person a day about your issue. A one-on-one plea will be much more effective in convincing theÂ listener. On the second day, there will be two of you who can approach two more people. On dayÂ three, there are four of you to approach four more people. On day five, the eight of you convinceÂ eight more people, and so on. By day 12, there are over 2,000 people who know about your cause,Â and by day 30, over one billion people are talking about the issue that is so close to your heart! YetÂ you personally talked to only 30 people. By the way, now you know why unfounded rumors spread

so quickly.

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