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

Example 2.

For what value of x is 5(x - 10) = x + 10?

· Memorize these six steps in order, and use this method whenever you have to solve this type of equation or inequality.

· When you have to solve for one variable in terms of others, treat all the others as if they were numbers, and apply the six-step method.

Absolute value, Radical, and fractional equations and inequalities.

Example 1. For what value of x is

Example 2. For what values of x is ?

[] ?[]?

Case 1: x > 0.

Case 2: x < 0.

Finally, by combining the two cases, we see that the solution set is {x| x < 0 or x > 16}.

Example 3. For what values of x is 3|x+5| - 5 = 7?

3|x+5| - 5 = 7 ? 3|x + 5| = 12? |x + 5| = 4 ? x + 5 = 4 or x + 5 = - 4 ? x = - 1 or x = - 9.

Quadratic Equations

A quadratic equation is an equation that can be written in the form a, where a, b, and c are any real numbers with a ? 0. Any number, x, that satisfies the equation is called a solution or a root of the equation.

Quadratic Formula

If a, b, and c are real numbers with a ? 0 and if , then

Recall that the symbol ± is read “plus or minus” and that is an abbreviation of the quadratic equation for or .

As you can see, a quadratic equation has two roots, both of which are determined by the quadratic formula.

The expression that appears under the square root symbol is called the discriminant of the quadratic equation.

Example 1. What are the roots of the equation ?

a = 1, b = -2, c = -15

and

D =

So

or

Example 2. What are the roots of the equation ?

First, rewrite the equation in the form :

Then a = 1, b = -10, c = 25

and

D =

So

or

Notice that since 10 + 0 = 10 and 10 - 0 = 10, the two roots are each equal to 5.

Example 3. Solve the equation :

a = 2, b = -4, c = -1

and

D =

So

or .

Example 4. Solve the equation :

a = 1, b = -2, c = 2

and D =

So

or

· If a, b, and c are rational numbers with a ? 0, if , and if , then

If , then the sum of the two roots is and the product of the two roots is .

Example. Find a quadratic equation for which the sum of the roots is 5 and the product of the roots is 5.

For simplicity, let a = 1. Then = 5?b = -5, and So the equation satisfies the given conditions.

Key words

Exponential Equations

There are two ways to handle equations of this type: use the laws of exponents or use logarithms.

There is a big difference between the equations and . The first equation is much easier to solve than the second if you recognize that 16 is a power of 2.

Example 1. For what value of x is ?

Example 2. For what value of x is ?

Since 15 is not a power of 2, you must use logarithms.

Example 3. If , what is the ratio of x to y?

Systems of linear equations

When the graphs of the equations are lines, the equations are called linear equations. A system of equations is a set of two or more equations involving two or more variables. A solution consists of a value for each variable that will simultaneously satisfy each equation.

Each of the equations 2x+y=13 and 3x-y=12 has infinitely many solutions. However, only one pair of numbers, x = 5 and y = 3, satisfies both equations simultaneously: 2(5) + 3 = 13 and 3(5) - 3 = 12. This then is the only solution of the system of equations:

There are three basic methods to solve a system of linear equations, such as the one above: two algebraic ones - the addition method and the substitution method - and one graphic one.

Addition method

Example 1.

The easiest way to solve the system of equations discussed above it to add the two equations:

Now solve for y by replacing x with 5 in either of the two original equations.

For example: 2(5) + y = 13 ? 10 + y = 13 ? y = 3

So, the unique solution is x = 5 and y = 3.

Example 2.

To solve the system , you cannot just add the two equations.

Fortunately, there is an easy remedy. If you multiply the first equation by 2, you will get an equivalent equation: 4x + 2y = 26. Now you can use the addition method to solve the new system:

Then, substitute 4 for x in one of the original equations:

2(4) + y = 13 ? 8 + y = 13 ? y = 5

So, the solution is x = 4 and y = 5.

Example 3.

To solve the system , multiply the first equation by 3 and the second equation by -2, and then add the new equations:

Now replace y by 7 in either of the original equations:

The solution is x=5 and y=7.

The substitution method

If in system of equations either variable has a coefficient of 1 or -1, solving the system by the substitution method may be just as easy or even easier than solving it by the addition method.

Now in the second equation you can replace y by 13-2x:

This is now a simple equation in one variable that you can solve using the six-step method.

Then substitute 5 for x:

The solution is x=5 and y=3.

The graphing method

System of linear equations can also be solved graphically. To solve

graph each of the lines and find the point where the two lines intersect. The x- and y-coordinates of the point of intersection are the x and y values of the solution.

To solve this problem, you can make a rough sketch.

Solving linear-quadratic systems

Example.

To solve the system , use the substitution method. Replace the y in the second equation by 2x - 1.

If x = 3, then y = 2(3) - 1 = 5; and if x = 1, then y = 2(1) - 1 = 1.

So, there are two solutions: x = 3, y = 5 and x = 1, y = 1.

Word problems

To solve word problems algebraically, you must treat algebra as a foreign language and translate “word for word” from English to algebra, just as you would from English into any foreign language. When translating into algebra, you should use some letter (often x) to represent the unknown quantity you are trying to determine. Review all of the examples in this section so that you master this translation process.

Rate problems

The basic formula used in all rate problems is

Of course, you can solve for one of the other variables:

Sometimes in rate problems, the word speed is used instead of rate. When you solve word problems, be sure you use consistent units.

Example 1.

John drove from his house to his office at an average speed of 30 miles per hour. If the trip took 40 minutes, how far, in miles, is it from John's house to his office?

Solution.

You know the rate (30mph) and the time (40 minutes), and you want to find the distance. Of course, you are going to use the formula d = rt. However, if you write d = (30)(40)=1200, you know something is wrong. Clearly, John didn't drive 1200 miles in less than one hour. The problem is that the units are wrong. The formula is really:

So, you have to convert 40 minutes to hours:

Then 60x = 40?. So it took John hours to get to his office.

Now you can see the formula d=rt, with r = 30 and t = .

hours) =

Example 2.

If Brian can paint a fence in 4 hours and Scott can paint the same fence in 6 hours, when working together, how long will it take the two of them to paint the fence?

Solution.

Call painting the fence “the job”. Then Brian works at the rate of . Similarly, Scott's rate of work is . Together they can complete ( jobs per hour. Finally,

So, it will take Brian and Scott to paint the fence.

Age problems

In age problems, it often helps to organize the given data in a table.

Example.

In 2001, Lior was four times as old as Ezra, and in 2003, Lior was twice as old as Ezra. How many years older than Ezra is Lior?

Solution.

Let x represent Ezra's age in 2001, and make the following table:

In 2003, Lior was twice as old as Ezra, so

So in 2001, Ezra was 1 and Lior was 4. Lior is 3 years older than Ezra.

Percent problems

Example 1.

There are twice as many girls as boys in a biology class. If 30% of the girls and 45% of the boys have completed a lab, what percent of the students in the class have not yet completed the lab?

Solution.

If x represents the number of boys in the class, then 2x represents the number of girls in the class. Then of the 3x students in the class, the number of students who have completed the lab is

The fraction of students who have completed the lab is

So, 65% of the students have not yet completed the lab.

Key words

Plane geometry

Lines and Angles

Lines are usually referred to by lowercase letters, such as l, m, and n. We can also name a line using two of the points on the line. If A and B are points on line l, we can refer to line l as a line .

represents the ray that consists of point A and all the points on

represents the line segment that consists of points A and B and all the points on that are between them.

Finally, represents the length of segment

If two line segments have the same length, we say they are congruent. The symbol is used to indicate congruence, so in the figure below we have .

Angles

An angle is formed by the intersection of two line segments, rays, or lines. The point of intersection is called the vertex of the angle.

· An acute angle measures less than .

· A right angle measures .

· An obtuse angle measures more than but less than .

An angle can be named by three points: a point on one side, the vertex, and a point on the other side, in that order. When there is no possible ambiguity, we can name the angle just by its vertex.

If two or more angles form a straight angle, the sum of their measures is

Example.

If in the figure below a:b:c=2:3:4, then a=2x, b=3x, c=4x. So,

2x+3x+4x=180 ? 9x = 180 ? x=

· The sum of the measures of all nonoverlapping angles around a point is

· Vertical angles have equal measures.

Perpendicular and parallel lines

Two lines that intersect to form right angles are called perpendicular. Two lines that never intersect are said to be parallel. Consequently, parallel lines form no angles, However, if a third line, called transversal, intersects a pair of parallel lines, eight angles are formed, and the relationship between these angles are very important.

If pair of parallel lines is cut by a transversal that is perpendicular to the parallel lines, all eight angles are right angles.

If a pair of parallel lenis is cut by a transversal that is not perpendicular to the parallel lines:

Four of the angles are acute, and four are obtuse.

All four acute angles are congruent.

All four obtuse angles are congruent.

The sum of the measures of any acute angle and any obtuse angle is

Angles d and f, as well as angles c and e, are called alternate interiors angles because they are interior to the two parallel lines and on alternate sides of the transversal.

Angles a and g, as well as angles b and h, are called alternate exterior angles because they are exterior to the parallel lines and on alternate sides of the transveral.

Pairs a and e, and f and h, and c and g are called corresponding angles because they are in corresponding positions in relationship to the parallel lines and transversal.

Pairs d and e as well as c and f are called consecutive interior angles because they are interior to the parallel lines and on the same of the transversal.

Triangles

Sides and angles of a triangle

In any triangle, the sum of the measures of the three angles is

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

Also, in any triangle:

· The longest side is opposite the largest angle.

· The shortest side is opposite the smallest angle.

· Sides with equal lengths are opposite angles with equal measures (the angles opposite congruent sides are congruent).

· A triangle is called scalene if the three sides all have different lengths. Then by previous fact, the three angles all have different measures.

· A triangle called isosceles if two sides are congruent.

· A triangle is called equilateral if all three sides are congruent. Since the sum of the measures of three angles is , each angle is

· Acute triangles are triangles in which all three angles are acute. An acute triangle could be scalene, isosceles, or equilateral.

· Obtuse triangles are triangles in which one angle is obtuse and two are acute. An obtuse triangle could be scalene or isosceles.

· Right triangles are triangles that have one right angle and two acute angles. A right triangle could be scalene or isosceles. The side opposite the angle is called the hypotenuse, it is the longest side. The other two sides are called the legs.

Right triangles

If a and b are the measures in degrees, of the acute angles of a right triangle, 90 + a + b = 180 ? a + b = 90.

In any right triangle, the sum of measures of the two acute angles is

Example.

To find the average of a and b in below, a + b = 90, so

Pythagorean theorems and corollaries

Let a, b and c be the lengths of the sides of , with a ? b ? c.

· if and only if angle C is a right angle.

· if and only if angle C is obtuse.

· if and only if angle C is acute.

For any positive number x, there is a right triangle whose sides are 3x, 4x, 5x.

For example:

x = 13, 4, 5x = 515, 20, 25

x = 26, 8, 10x = 1030, 40, 50

x = 39, 12, 15x = 50150, 200, 250

x = 412, 16, 20x = 100300, 400, 500

Other right triangles with integer sides that you should recognize immediately are the ones whose sides are 5, 12, 13 and 8, 15, 17. These sets of three numbers are often referred to as Pythagorean triples.