Дипломная работа: Методические основы подготовки будущих учителей математики в условиях полиязычного образования

For any real numbers a, b, c, and d where d :

a(b + c)=ab + aca(b - c) = ab - ac

Key words

Fractions, Decimals and Percents

Numbers such as in which one integer is written over a second integer called fractions.

The center line is called the fraction bar. The integer above the fraction bar is called numerator, and the integer below the fraction bar is called the denominator.

When a number is actually written as we can call it a fraction.

For example, of 4, 0.7, 0.222, …20%, and , only is a fraction.

Numbers that cannot be expressed as fractions are called irrational numbers. Any nonterminating, nonrepeating decimal is an irrational number.

A fraction is in lowest terms if no single positive integer greater than 1 is a factor of both the numerator and denominator. For example, is in lowest terms since no integer greater than 1 is a factor of both 8 and 15; but is not in lowest terms since 2 is a factor of both 8 and 18.

Every fraction can be reduced to lowest terms by dividing the numerator and the denominator by their greatest common factor (GCF). If the GCF is 1, the fraction is already in lowest terms.

Every fraction can be expressed as a decimal (or a whole nimber) by dividing the numerator by the denominator.

· If a fraction is written in lowest terms and if the only prime factors of the denominator are 2 or 5, the decimal terminates.

· If a fraction is written in lowest terms and if the denominator has any prime factor other than 2 or 5, the decimal repeats.

Example 1.

Since 4, 5, 8, 10, 16, 20, 25, and 40 have no prime factors other than 2 and 5, the decimal equivalents of each of the following fractions terminate:

Example 2.

Since 6, 7, 9, 12 and 22 all have prime factors other than 2 and 5 (6, 9 and 12 are multiples of 3; 7 is a multiple of 7, and 22 is a multiple of 11), the decimal equivalents of each of the following fractions repeat:

….

….

….

….

To complete and , cross multiply.

If ad =bc, then

If ad > bc, then

If ad < bc, then

Operations with Fractions

To multiply two fractions, multiply their numerators and multiply their denominators.

=

To multiply a fraction by any other number, write that number as a fraction whose denominator is 1:

Before multiplying fractions, reduce. You may reduce by dividing any numerator and any denominator by a common factor.

When a problem requires you to find a fraction of a number, multiply. Since a percent is just a fraction whose denominator is 100, you also multiply to find a percent of a number.

Example.

If of the 840 students at Monroe High School are freshmen and if 30% of the freshmen play musical instruments, how many freshmen play musical instruments?

There are freshmen. Of these, 30% play an instrument:

30% of 240=

The reciprocal of any nonzero number, x, is the number . The reciprocal of the fraction is the fraction .

To divide any number by a fraction, multiply that number by the reciprocal of the fraction.

20ч

To add or subtract fractions with the same denominator, add or subtract the numerators and keep the denominator.

and

To add or subtract fractions with different denominators, first rewrite the fractions as equivalent fractions with the same denominators.

If is the fraction of the whole that satisfies some property, then is the fraction of the whole that does not satisfy it.

Example.

In a jar, of the marbles are red, are white, and are blue. What fraction of the marbles are neither red, white nor blue?

Solution.

The red, white and blue marbles constitute

of the total

So, of the marbles are neither red, white, nor blue.

Key words.

Operations with mixed numbers

A mixed numbers is a number such as 3 that consists of an integer followed by a fraction. It is an abbreviation for the sum of the integer and the fraction; so 3.

Complex fractions

A complex fraction is a fraction, such as or , that has a fraction in its numerator, denominator, or both.

A complex fraction can be satisfied in two ways: (i) multiply every term in the numerator and denominator by the least common multiple of all the denominators that appear in the fraction or (ii) simplify the numerator and the denominator and then divide.

Example.

To simplify , multiply each term by 12, the LCM of 6 and 4:

or write

Percents

As mentioned previously, the word percent means hundredth. We use the symbol % to express the word “percent”. For example, “23 percent” means “23 hundredths” and can be written with a % symbol, as a fraction, or as a decimal:

To convert a percent to a decimal, drop the % symbol and move the decimal point two places to the left, adding 0's if necessary.

To convert a percent to a fraction, drop the % symbol, write the number over 100, and reduce.

20% = 0.20 = =

100% = 1.00 = =

37.5% = 0.375 = =

3% = 0.03 =

Percent increase and decrease

The percent increase of a quantity is:

The percent decrease of a quantity is:

Example.

From 1980 to 1990, the population of a town increased from 12000 to 15000. Since the actual increase in the population was 3000, percent increase in the population was

An increase of a% followed by an increase of b% always results in a larger increase than a single increase of (a+b)%. Similarly, a decrease of a% followed by a decrease of b% always results in a smaller decrease than a single decrease of (a+b)%.

To increase a number by r%, multiply it by (1+r%). To decrease a number by r%, multiply it by (1 - r%).

Example.

Starting on January 1, 1990, the population of Centerville increased by 5% every year. If Centerville's population was 10000 on January 1, 1990, what was its population on January 1, 1992

Since the population on January 1, 1990 was 10000, to find the population one year later on January 1, 1991, multiply by 1.05:

(1+5%)(10000) = (1+0.05)(10000) = (1.05)(10000) = 10500

To get the population on January 1, 1992, again multiply by 1.05.

(1.05)(10500) = 11025

Note that (1.05)(10500) = (1.05)[(1.05)(10000)] = (1.05(10000).

This process can be continued for any numbers of years.

If an initial quantity A increases r% per year, then the amount at the end of t years is given by A(t) = A(1+r%

Example.

If the population of Centerville (which was 10000 on January 1, 1990) grew at a rate of 5% per year for 10 years, what was its population on January 1, 2000?

The population after 10 years was

A(10) = 10000(1 + 0.05 = 10000(1.05=10.

Note that 10 increases of 5% resulted not in a 50% increase, but in a total increase of 6289 or nearly 63%. Also note that the population on January 1, 2000 was 163% of the population on January 1, 1990. Similarly, if the area of one square is 3 and the area of a second square is 18, the area of the larger square is 600% of the area of the smaller square:

600% of 3 =

However, the area of the large square is 500% greater than the area of the small square. The increase in the area from the small square to the large one is 15, and,

.

Ratios and Proportions

A ratio is a fraction that compares two quantities that are measured in the same units. The first quantity is the numerator, and the second quantity is the denominator.

For example, if in right , the length of leg is 6 inches and the length of leg is 8 inches, we say that the ratio of AC and BC is 6 to 8, which is often written as 6: but is just the fraction . Like any fraction, a ratio can be reduced and can be converted to a decimal or a percent.

AC to BC = 6 to 8 = 6: 8 = .

AC to BC = 3 to 4 = 3: 4 = .

If you know that AC = 6 inches and BC = 8 inches, you know that the ratio of AC to BC is 6 to 8. However, if you know that the ratio of AC to BC is 6 to 8, you cannot determine how long either side is. They may be 6 and 8 inches long but not necessarily.

Their lengths, in inches, may be 60 and 80 or 300 and 400 since and are both equivalent to the ratio . In fact, there are infinitely many possibilities for the lengths.

The important thing to observe is that the length of can be any multiple of 3 as long as the length of is the same multiple of 4.

· If two numbers are in the ratio of a: b, then for some number x, the first number is ax and the second number is bx.

· In any ratio problem, write x after each number and use some given information to solve for x.

Example.

In right triangle, the ratio of the length of the shorter leg to the length of the longer leg is 5 to 12. If the length of the hypotenuse is 65, what is the perimeter of the Triangle?

Solution.

Draw a right triangle and label it with the given information; then use the Pythagorean theorem.

So AC = 5(5) = 25, BC = 12(5) = 60, and the perimeter equals 25 + 60 + 65 = 150. Ratios can be extended to 3 or 4 or more terms. For example, we can say that the ratio of freshmen to sophomores to juniors to seniors in a school band is 3: 4: 5: 5. This means that for every 3 freshmen in the band there are 4 sophomores, 5 juniors, and 4 seniors.

Key words

Proportions

A proportion is an equation that states that two ratios are equivalent. Since ratios are just fractions, any equation such as , in which each side is a single fraction, is a proportion.

You can solve proportions by cross multiplying. If , then ad = bc.

Example.

If , then 20(x+3) = 19(x+5) ?20x+60=19x+95? x=35

A rate is a fraction that compares two quantities measured in different units. Rates often use the word “per” as in miles per hour and dollars per week.

· Set up rate problems just like ratio problems.

· Solve the proportions by cross multiplying.

Example.

Frank can type 600 words in 15 minutes. If Diane can type twice as fast, how many words can she type in 40 minutes?

Solution.

Since Diane types twice as fast as Frank, she can type 1200 words in 15 minutes. How handle this rate problem exactly as you would a ratio problem. Set up a proportion and cross multiply:

15x = (40)(1200) = 48000 ? x = 3200

ALGEBRA

Polynomials

A monomial is aa number or a variable or a product of numbers and variables. Each of the following is a monomial:

The numerical portion of a monomial is called the coefficient and is always written in front of the variables. The coefficient of is 7. If there is no number in front of a variable, the coefficient is 1 or -1, because x means 1x and -xy means -1xy.

A polynomial is a monomial or the sum of two or more monomials. Each monomial that makes up a polynomial is called a term of the polynomial. The terms of a polynomial are separated by addition signs (+) and subtraction signs (-). Each of the following is a monomial:

Note that and so is the sum of the monomials and - 7. Also, the sum, difference, and product of any polynomials is itself a polynomial.

The first polynomial in the preceding list ( is a monomial because it has one term. The second (, third (, fifth ( , and sixth () polynomials are called binomials because they have two terms. The fourth ( and seventh () polynomials are called trinomials because they have tree terms.

Example.

To evaluate when x = -2 and y = , rewrite the polynomial, replacing each x by -2 and each y by . Be sure to write each number in parentheses.

· The only terms of a polynomials that can be combined are like terms.

· To add two polynomials, write each in parentheses and put a plus sign between. Then erase the parentheses and combine like terms.

· To subtract two polynomials, write each one in parentheses and put a minus sign between them. Then change the minus sign to a plus sign, change the sign of every term in the second parentheses, and use previous fact to add them.

· To multiply monomials, first multiply the coefficients, and then multiply their variables (one by one) by adding their exponents.

· To multiply a polynomial by a monomial, just multiply each term of the polynomial by the monomial.

· To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial and simplify by combining terms, if possible.

· To divide a polynomial by a monomial, divide each of term by the monomial. Then simplify each term by reducing the fractions formed by the coefficients to lowest terms and applying the laws of exponents to the variables.

· The first step in factoring a polynomial is to look for the greatest common factor of all the terms and, if there is one, to use the distributive property to remove it.

To factor a trinomial, remove a common factor, if there is one, and then use trial and error to find the two binomials whose product is that trinomial.

Example.

Algebraic fractions

Although the coefficient of any term in a polynomial can be a fraction, such as , the variable itself cannot be in the denominator. Expressions such as and , which do have variables in their denominators, are called algebraic fractions. You will have no trouble manipulating algebraic fractions if you treat them as regular fractions, using all the standard rules for adding, subtracting, multiplying, and dividing.

Whenever you have to simplify an algebraic fraction, factor the numerator and denominator, and divide out any common factors.

Example. Simplify .

Solution.

Note that is an identity. For every real number (except -2 and -3, for wgich the original fraction is undefined), the expressions have the same value. For example, when x = 8:

and

Similarly, when x = 3, both expressions equal 0, and when x = - 4, both expressions equal 7.

Equations and Inequalities

Example 1.

The following solution of the equation illustrates each of the six steps.