Vocabulary 4th by Paqui Cabrera

MATHS VOCABULARY AND ACTIVITIES

CSE

Bilingual Program

Bilingual Program MATHS UNIT 1 – REAL NUMBERS 

Natural Numbers (N): The numbers used for counting. That is, the numbers 1, 2, 3, 4, etc.



Whole Numbers or Nonnegative Integers: The numbers 0, 1, 2, 3, 4, 5, etc.



Integers (Z): All positive and negative whole numbers (including zero). That is, the set: Z = {... , –3, –2, –1, 0, 1, 2, 3, ...}.



Rational Numbers (Q): All positive and negative fractions, including integers and improper fractions. Formally, rational numbers are the set of all real numbers that can be written as a ratio of integers with nonzero denominator.



Irrational Numbers (I): Real numbers that are not rational. Irrational numbers include numbers such as



2  29 , 4

, π, e, etc.

Real Numbers(R): All numbers on the number line. This includes (but is not limited to) positives and negatives, integers and rational numbers, square roots, cube roots , π (pi), etc.



Rounding a Number: A method of approximating a number using a nearby number at a given degree of accuracy. For example, 3.14159265... rounded to the nearest thousandth is 3.142. That is because the third number after the decimal point is the thousandths place, and because 3.14159265... is closer to 3.142 than 3.141.



Truncating a Number: A method of approximating a decimal number by dropping all decimal places past a certain point without rounding. For example, 3.14159265... can be truncated to 3.1415. Note: If 3.14159265...were rounded to the same decimal place, the approximation would be 3.1416



Interval: The set of all real numbers between two given numbers.

Bilingual Program MATHS The two numbers on the ends are the endpoints. The endpoints might or might not be included in the interval depending whether the interval is open, closed, or half-open (same as half-closed)

 Closed Interval: An interval that contains its endpoints  Open Interval: An interval that does not contain its endpoints.  Half-Closed Interval or Half-Open Interval: An interval that contains one endpoint but not the other.

There are three main ways to show intervals: Inequalities, The Number Line and Interval [a, b) (a, b]

a≤x<b a<x≤b

closed on left, open on right open on left, closed on right

 Union and Intersection: Example: x ≤ 2 or x >3 . On the number line it would look like this:

(-∞, 2] U (3, +∞) We used a "U" to mean Union (the joining together of two sets). There is also "Intersection" ∩, which means "has to be in both". Think "where do they overlap?" Example: (-∞, 6] ∩ (1, ∞). The Intersection (or overlap) of those two sets goes from 1 to 6: (1, 6]

Bilingual Program MATHS  Reading operations: o 3+5=8

Three plus five equals eight

o 12 - 3 = 9 Twelve minus seven equals nine ( Subtract 3 from 12 ) o 8 x 4 = 32

Eight times four equals thirty-two ( or eight multiplied by four equals thirty-

two) o 28 : 4 = 7

Twenty-eight divided by two equals seven ( or eight divided into two equals

seven)

 Reading of fractions:

1 2

17 2

= a half

3 4

= three fourths / three quarters

= seventeen halves

= nine eighths

21 = twenty-one over two hundred and sixty-five 265

 Reading of decimal numbers: o o o o o

9 8

1.827 = one point eight two seven 35.15 = thirty-five point one five 3. 1414…= three point one four repeating 3.14343….= three point one four three with four three repeating 3.01111… = Three point zero one with one repeating

Bilingual Program MATHS

UNIT 2– POWERS AND ROOTS

POWERS AND ROOTS:      

power base index, indices square number square root Radical: The



nth Root: The number that must be multiplied times itself n times to equal a given value.

symbol, which is used to indicate square roots or nth roots.

The nth root of x is written

Notes: When n = 2 a nth root is called a square

root. 

Radicand: the number inside the radical symbol.  Reading powers :

     

42     four squared. To square a number is to multiply it by itself. 7 3     seven cubed 154     fifteen to the power of four 65289     sixty  five to the power of two hundredand eighty nine Write as a single power You have to calculate the square of this /You have to square this.--- Tienes que elevar al cuadrado esto

 Reading roots: 25  5    square root of twenty  five is five

  5

32  2     the fifth root of thirty  two is 2

     the sixth root of twenty  six is ...



You have to calculate the square root of .../You have to take the square root of … ---Tienes que sacar raíz cuadrada de …



Calculate the cube root of… /Take the cube root of… --- Saca la raíz cúbica de…

Bilingual Program MATHS POWERS WITH INTEGER EXPONENT: ď&#x192;&#x2DC; A power is a product of equal factors:

a.a.a.a.aâ&#x20AC;Ś.a = an

ď&#x192;&#x2DC; The base(a) is the number that you multiply; the exponent (n) shows how many times you have to multiply the base ď&#x192;&#x2DC; To multiply powers with the same base, you have to write the same base and add the exponents an. am = an+m ď&#x192;&#x2DC; To divide powers with the same base, you have to write the same base and subtract the exponents an: am = an â&#x20AC;&#x201C; m ď&#x192;&#x2DC; To find the power of another power, you have to write the base and multiply the exponents (an)m = an.m SCIENTIFIC NOTATION OR STANDARD FORM: Scientific notation is a product of a number between 1 and 10 and a power of 10 aÂˇ 10n (1 ď&#x201A;Ł a ď&#x20AC;ź 10)

ROOT OF A NUMBER đ?&#x2019;?

â&#x2C6;&#x161;đ?&#x2019;&#x201A; = đ?&#x2019;&#x192; đ?&#x2019;&#x160;đ?&#x2019;&#x2021; đ?&#x2019;&#x192;đ?&#x2019;? = đ?&#x2019;&#x201A;

Ex: â&#x2C6;&#x161;đ?&#x;?đ?&#x;&#x201C;

Âąđ?&#x;&#x201C;

ď&#x192;Ź ď&#x192;Ż ď&#x20AC;˝ square root of â&#x2C6;&#x161;đ?&#x;?đ?&#x;&#x201C; is a square-root radical ď&#x192; ď&#x192;Ż ď&#x192;Ž25 ď&#x20AC;˝ radicand

NUMBER OF ROOTS OF ONE RADICAL The number of roots of a radical is: ď&#x192;&#x2DC; Two: if the index is an even number and the radicand is positive 16 ď&#x20AC;˝ ď&#x201A;ą4 ď&#x192;&#x2DC;

One: if the index is an odd number, if radicand is positive, the root is positive and negative if the radicand is negative

ď&#x192;&#x2DC;

ď&#x20AC; 27 ď&#x20AC;˝ ď&#x20AC;3 ,

27 ď&#x20AC;˝ 3

Nothing: If the index is an even number and the radicand is negative

ď&#x20AC; 16

FRACTIONAL POWERS m

a m ď&#x20AC;˝ a n or

a ď&#x20AC;˝ an

Fractional powers are useful when we need to calculate roots using a scientific calculator

Bilingual Program MATHS PROPERTIES OF RADICALS Let “a” and “b” be real numbers, variables, and let m and n be positive integers, then:  Product of radical with the same index : To multiply two radicals with the same index , write the same index and multiply the radicands n a  n b  n a  b  Division of roots with the same index: To divide two radicals with the same

index, divide

a n a  b0 n b b  Power of a root: To calculate the power of a root write the same index and raise its radicand the radicands and write the same index

 a n

at this power

 n am

 Root of a root: To calculate the root of a root, multiply the index and leave the same radicand.

a  mn a

ADDITTION AND SUBTRACTION OF RADICALS You can only add or subtract radicals together if they are like radicals. In this case, we add or subtract the terms in front of each like radical. Remember! Like radicals are radicals that have the same root number and radicand.

Simplify: Step 1: Simplify each radical. Step 2: Add or subtract the radicals. Remember that we can only combine like radicals. RATIONALISING Rationalising an expression means getting rid of any surds from the bottom of fractions. Usually when you are asked to simplify an expression it means you should also rationalise it.

Bilingual Program MATHS UNIT 3 - ALGEBRAIC EXPRESSIONS 

Expressions containing letters and numbers are called algebraic expressions.



If we substitute letters for numbers in an algebraic expression and carry out the operation indicated we get a number which is its numerical value of the algebraic expression for the values of the given letters.

 A monomial is an algebraic expression where the only operations indicated with the letters are products and powers where the exponent is a natural number. Examples of monomials are: 

5 y 3 , z, 3a 2 b

Similar monomials are those which contain the same letters with the same exponents.

 The degree of a monomial is the sum of the exponents on the variables contained in it. For example, the degree of 2a 3b would be 3 + 1 = 4 and the degree of 5y 3 would be 3.



A coefficient is the numeric factor of your monomial. For instance: 5 is the coefficient in 5y 3 .



A constant term is a monomial that contains only a number. In other words, there is no variable in a constant term.



A polynomial is a finite sum of monomials where the exponents on the variables are non-negative integers. The monomials in a polynomial are called terms.



The degree of the polynomial is the largest degree of all its terms.



The terms are like terms when the monomials are similar. In this case we collect like terms.

For example: After collecting like terms in the expression: 3x 2  4x3  2x3  x  9  5x  3 , we will write it in this way:  2x3  3x 2  4x  6 . This is a third- degree polynomial. And the polynomial

is a fifth-degree polynomial.

Bilingual Program MATHS In the Polynomial: P( x)  an x n  an1 x n1  ....  a1 x  a0

a n is the leading coefficient a 0 is the constant term n is the degree of the polynomial P(x) is a nth-degree polynomial



This polynomial is written in descending order (this happens when the term that has the highest degree is written first, the term with the next highest degree is written next, and so forth).

Ruffini's rule: It allows the rapid division of any polynomial by a binomial of the form x − c. Example: Let: P( x)  2x 3  3x 2  4

and

Q( x)  x  1

1) Write down the coefficients and c. Note that, as P(x) didn't contain a coefficient for x, we've written 0: | 2 3 0 -4 | -1 | ----|------------------------|

2) Pass the first coefficient down: | 2 3 0 -4 | -1 | ----|------------------------| 2

3) Multiply the last obtained value by c: | 2 3 0 -4 | -1 | -2 ----|-------------------------| 2

Bilingual Program MATHS 4) Add the values: | 2 3 0 -4 | -1 | -2 ----|------------------------| 2 1

5) Repeat steps 3 and 4 until we've finished: | 2 3 0 -4 | -1 | -2 -1 1 ----|---------------------------| 2 1 -1 -3 = R (remainder) {quotient coefficients}

So, as dividend= divisor × quotient + remainder, then: P( x)  Q( x)  S ( x)  R , where S ( x)  2x 2  x 1 and R = -3

 The Remainder Theorem: When you divide a polynomial P(x) by x-c, the remainder R will be P(c). P(c) = R

 Root of a polynomial: A number x = c is called a root of the polynomial P(x) if P(c) = 0 Note that if c is a root, then x-c is a factor (and vice versa).

 What is factoring? If you write a polynomial as the product of two or more polynomials, you have factored the polynomial.





Example: When we write x3  3x 2  2x  6  x  3  x 2  2 , we have factored the polynomial; x  3, x 2  2 are called factors of the polynomial and 3, 2 ,  2 are the roots of this polynomial.

Bilingual Program MATHS ALGEBRAIC EXPRESSIONS -- Activities 1) Answer true or false. Reason your answers. 4 a) ab and

1 4 ab are similar monomials. 5

b) 7a 2 b 4 is a fourth-degree monomial. c) 2x 2 y is a monomial. 3 d) Given the polynomial  x  3x  9 :

    

It is a third-degree polynomial It is complete It is written in ascending order Its constant term is nine. Its leading coefficient is -1

e) The algebraic expression x 3 y  2xy 2  7xy  y  7 is a polynomial. f) 1 is a root of x 2  1 g) If the numerical value of a polynomial for a value -m is zero, the polynomial is divisible by x + m 2) Complete the following sentences: x + 4 is a ____________ of x 2  16 1 is a ____________ of x 2  1 If Q(3) = 0, the number 3 is a root of _________ If it is possible to take x as a common factor in a polynomial, _______ is a root of the polynomial. e) If P(5) = 0, ________is a factor of the polynomial. a) b) c) d)

3) Write   

a sixth-degree polynomial verifying all the following conditions: Its leading coefficient is 5 It doesn’t have a constant term 7 and -4 are roots of the polynomial

4) Write three equivalent algebraic fractions to the given:

x 2  3x x3 1 b) 3 x  2x  4 a)

5) Simplify the following algebraic fraction:

x 3  3x 2  3x  1 x 4  4x 3  6x 2  4x  1

Bilingual Program MATHS EXERCISES

--- POLYNOMIALS

1) Using algebraic identities, calculate:

𝑑) (𝑥 2 − 3) ∙ (𝑥 2 + 3)

c) ( 𝑥 4 − 𝑥)

𝑏) (2𝑥 + 3)2

𝑎) (3 + 𝑥) ∙ (3 − 𝑥)

𝑒) (5 𝑥 2 + 2𝑥) ∙ (5 𝑥 2 − 2𝑥)

2) Factor the following polynomials using algebraic identities: ALGEBRAIC IDENTITIES 2

𝑎) 𝑥 − 4

𝑏)𝑥 + 4𝑥 + 4

𝑑) 2𝑥 2 − 50

𝑒)

𝑔) 𝑥 2 − 49𝑥

ℎ) 𝑥 2 − 49

𝑖) 𝑥 3 − 49𝑥

𝑗) 6𝑥 4 − 150𝑥 2

𝑘) 𝑥 2 − 14𝑥 + 49

𝑙) 𝑥 3 − 𝑥

𝑚) 2𝑥 3 − 28𝑥 2 + 98𝑥

𝑛) 5𝑥 5 − 80𝑥

𝑜) 121 − 𝑥 2

𝑝) 𝑥 4 − 18𝑥 2 + 81

𝑞) 5𝑥 5 − 10𝑥 4 + 5𝑥 3

𝑟) 4𝑥 4 −

𝑠)9𝑥 4 + 1 − 6𝑥 2

𝑐) 4𝑥 − 9

1 2 𝑥 + 9 − 2𝑥 9

𝑡) 𝑥 4 − 2 𝑥 2 + 16

𝑓) 4𝑥 2 − 𝑦 2

81 256

𝑢) 4𝑥 + 𝑥 2 + 4

𝑣) 𝑥 3 − 6𝑥 2 + 9𝑥

3) Factor the following polynomials:

𝑎) 𝑥 4 − 6𝑥 3 − 3𝑥 2 + 52𝑥 − 60 𝑑) 𝑥 3 + 𝑥 2 − 9𝑥 − 9 𝑔) 𝑥 3 + 6𝑥 2 + 11𝑥 + 6

𝑏) 𝑥 3 − 6𝑥 2 + 9𝑥 𝑒) 𝑥 4 − 2𝑥 2 + 1 ℎ) 𝑥 2 − 8𝑥 + 7

𝑐) 𝑥 3 − 7𝑥 2 + 15𝑥 + 9 𝑓) 𝑥 4 + 6𝑥 3 + 4𝑥 2 − 6𝑥 − 5 𝑖) 𝑥 3 + 2𝑥 2 − 13𝑥 + 10

4) Without dividing,

a) calculate the remainder of the following divisions: 𝑖) 𝑝(𝑥) = 𝑥 2 + 5𝑥 − 3 𝑒𝑛𝑡𝑟𝑒 𝑥 + 3

𝑖𝑖) 𝑞(𝑥) = 𝑥 2 + 𝑥 + 1 𝑒𝑛𝑡𝑟𝑒 𝑥 −

1 2

b) answer if the following polynomials are divisible by x + 1: 1 3 1 𝑖) 𝑝(𝑥) = 4𝑥 2 − 3𝑥 + 2 𝑖𝑖) 𝑞(𝑥) = + 𝑥 − 2𝑥 2 + 𝑥 3 2 5 3 5) Calculate the value of the letter m so that the polynomial p(x) can verify the given condition: 𝑎) 𝑝(𝑥) = 𝑥 3 − 𝑥 2 + 𝑚𝑥 − 1 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥 + 1 𝑏) 1 𝑖𝑠 𝑎 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑝(𝑥) = 𝑥 3 − 𝑥 2 + 𝑚𝑥 − 1 𝑐) 𝑇ℎ𝑒 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑜𝑓 𝑝(𝑥) = 𝑚𝑥 2 + 2𝑚𝑥 − 1 𝑖𝑛𝑡𝑜 𝑥 − 2 𝑖𝑠 2 𝑑) 𝑝(𝑥) = 𝑥 4 + 3𝑥 3 − 2𝑥 2 + 𝑥 − 𝑚 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥 + 3

Bilingual Program MATHS UNIT 4 – EQUATIONS AND INEQUALITIES Igualdad

= equality

Identidad

= identity

Ecuación

= equation

Inecuación

= inequality

Ecuación de primer grado

= a linear equation / a first- degree equation

Ecuación de segundo grado

= a quadratic equation / a second-degree equation

Ecuación de grado 3

= a cubic equation / a third-degree equation

Incógnita

= unknown

Solución

= solution

Ecuaciones equivalentes

= equivalent equations

Ecuaciones con paréntesis y denominadores

= equations involving brackets and fractions

Quitar denominadores

= cancel fractions

Quitar paréntesis

= multiply out brackets / remove brackets

Multiplicar ambos miembros por…

= multiply both sides by ….

Restar 3 en ambos miembros

= subtract 3 from both sides

Pasar un término de un miembro a otro

= transfer a term from one side to another

EQUATIONS: An equation is a mathematical statement that has two algebraic expressions separated by an equal sign. The expression on the left side of the equal sign is the left-hand side of the equation, and it has the same value as the expression on the right side (right-hand side). The degree of an equation that has not more than one variable in each term is the exponent of the highest power to which that variable is raised in the equation.  LINEAR EQUATIONS: The highest exponent of a linear equation is 1. The standard form is: ax + b = c, where a,b,c are constants (numbers). Steps to solve linear equations: 1. If parentheses occur, multiply to remove them. 2. Use the addition/subtraction property to get all terms with a variable on one side and all numbers on the other side. 3. Collect like terms. 4. Apply the multiplication/division property to solve for the variable. 5. Verify the solution.

Bilingual Program MATHS 

QUADRATIC EQUATIONS:

They are equations that can be written in the form ax2 + bx + c = 0, where a, b and c are numbers, a  0 The quadratic formula is a general way of solving any quadratic equation:

 b  b 2  4ac x= 2a 

BIQUADRATIC EQUATIONS:

They are quartic equations with no odd-degree terms. They can be written in the form:

ax4  bx2  c  0 , a  0 Solving biquadratic equations: Change x 2 = t , x 4 = t 2 ; this generates a quadratic equation with the unknown, t : For every positive value of t there are t w o v al u es of x , find:

x t

Solve: x 4 13x 2  36  0

E xa m p l e :

We change: x 2  t  t 2 13t  36  0 So: x 2  9



at2 + bt + c = 0

 x  9  x  3 and

t

13  169  144 13  5  2 2

 t=9 y t=4

x 2  4  x  4  x  2 .

EQUATIONS THAT CAN BE SOLVED FACTORISING

E xa m p l e s: 1) 7(x – 3)(2x + 1)(x + 5) = 0. We find the solutions by solving every of the brackets In this case the solutions are:

x = 3; x = 

1 ; x= -5 2

1 3

2) 3 x 5  x 4  9x 3  9x 2  2x  0  x  1; x  2; x   ; x  0

As the roots of the polynomial are the solutions of the equation, we need to find the roots. Therefore, in this kind of equations we work similar to when we factorise polynomials: 1. If you can take common factor, do it! 2. Use Ruffini’s rule to find whole roots of the polynomials 3. Solve the quadratic equation.

Bilingual Program MATHS 

RATIONAL EQUATIONS:

A rational expression is a fraction with a polynomial in the numerator and denominator. If you have an equation containing rational expressions, you have a rational equation. For example:

x 3x  5 x 1 x 1

TO SOLVE RATIONAL EQUATIONS, we use the following steps: 1. Find the Lowest Common Multiple (LCM) 2. Write the equation with the same denominator in both sides, arranging the numerators. 3. At this point, the two sides of the equation will be equal as long as the numerators are equal. 4. Carry out the equation that we get writing the numerators. 5. Check if the solutions are right. They mustn’t cancel anyone of the denominators. 

IRRATIONAL EQUATIONS:

Irrational equations or radical equations have the unknown under the radical.

x 2  2x  x  2x  2 is an irrational equation.

For example:

TO SOLVE AN IRRATIONAL EQUATION, follow these steps: 1. Isolate a radical in one of the sides 2. Square both members. 3. Solve the equation obtained. 4. Check if the solutions obtained verify the initial equation. 5. If the equation has several radicals, repeat the first two steps of the process to remove all of them. Example: Solve the following irrational equation

x 2  2x  x  2x  2 x 2  2x  2x  2  x

x

 2x

   x  2 2

 

x 2  2x  x  2 x 2  2x  x 2  4x  4  2x  4  x  2

Now we have to check if the solution obtained verifies the initial equation.

22  2  2  2  2  2  2 ? 02  42 ? 2  2  x  2 is the solution

Bilingual Program MATHS INEQUALITIES: An inequality is a mathematical statement that has two algebraic expressions separated by <, >, ≤, or ≥ To solve an inequality is to find all values of the variable that make the inequality true. Each of these numbers is a solution of the inequality, and the set of all such solutions is its solution set. Inequalities that have the same solution set are called equivalent inequalities. 

LINEAR INEQUALITIES IN ONE VARIABLE:

Solving linear inequalities is very similar to solving linear equations, except for one small but important detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative. Examples:

x3 2

x  2  3  x  1 Graphically, the solution is:

2 x  0  x  2  x  2

The only difference between the linear equation "x + 3 = 2" and this linear inequality is that I have a "less than" sign, instead of an "equals" sign. Note that the solution to a "less than, but not equal to" inequality is graphed with a parentheses (or else an open dot) at the endpoint, indicating that the endpoint is not included within the solution.

The only difference between the linear equation "2 – x = 0" and this linear inequality is the "greater than" sign in place of an "equals" sign.

or : 2  x, that is the same Graphically, the solution is:

Note that "x" in the solution does not "have" to be on the left. However, it is often easier to picture what the solution means with the variable on the left. Don't be afraid to rearrange things to suit your taste.

The solution method here is to divide both sides by a positive two. Graphically, the solution is:

4)  2x  4 Be careful:

x  2

This is the special case noted above. When I divided by the negative two, I had to flip the inequality sign.

Graphically, the solution is:

Bilingual Program MATHS 

QUADRATIC INEQUALITIES:

A quadratic inequality can be written in the form ax2 + bx + c ? 0, where ? is <, >, ≤, or ≥. Examples: 3x2 - 2x - 5 >0, x2 –x – 12 > 0.

The trick to solving a quadratic inequality is: 1º) Replace the inequality symbol with an equal sign and solve the resulting equation. The solutions to the equation will allow you to establish intervals that will let you solve the inequality. 2º) Plot the solutions on a number line creating the intervals for investigation. Pick a number from each interval and test it in the original inequality. 3º) If the result is true, that interval is a solution to the inequality.

For example: Solve x2 –x – 12 > 0. 1º) Change the inequality to = and solve the quadratic equation: x 2  x  12  0  x  4, x  3

2º)

3º) So the solution is:

x  3 or x  4

The solution written in interval notation is:

x   ,3  4,



RATIONAL INEQUALITIES : The process to solving them is very similar to solving quadratic inequalities.

Example:

x3 0 x 1

Steps to solve linear equations: 1. Find the number that makes the numerator equals 0 2. Find the number that makes the denominator equals 0 3. Plot the solutions on a number line creating the intervals for investigation. 4. Pick a number from each interval and test it in the original inequality. If the result is true, that interval is solution.

Bilingual Program MATHS To the previous example, the solution is: (,3]  (1,) Note: - the interval 1, is open on the left because 1 makes the denominator zero. We mustn’t divide by 0! - Infinite never is closed. You never can reach it!



LINEAR INEQUALITIES IN TWO VARIABLES:

The solution of a linear inequality in two variables like Ax + By > C is an ordered pair (x, y) that produces a true statement when the values of x and y are substituted into the inequality. A linear inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality. The half-plane that is a solution to the inequality is usually shaded.

Answer: Given the inequality 2x + 3y > 1, is (1, 2) a solution?

Example: Graph the solution to y < 2x + 3. 1º) Change the inequality sign for the equal sign, and graph the straight line y = 2x + 3:

2º) I need to shade one side of the line or the other. To decide it you have to try with one point (what you want) in one of the sides. If it verifies the inequality, that part is the solution; if it doesn’t, the solution is the other side.

3º) Finally, the side I shaded is the "solution region"

Bilingual Program MATHS QUESTIONS AND WORD PROBLEMS WITH EQUATIONS 1.- Fill in the gaps with the words given bellow. Rules to remember when writing algebraic equations 1. ______ through the entire problem. 2. Highlight the important information and ______ words that you need to solve the problem. 3. Identify the unknown, which is your ______________. 4. Look for key words that will help you to write the equation. Highlight the key words and write an ____________ to match the problem. 5. Write the equation or inequality. 6. Solve. 7. Write your answer in a _____________sentence. 8. ______ or justify your answer. key

variable

Check

Read

equation

complete

2.- Write in order the steps for solving a quadratic equation using the quadratic formula - Simplify the coefficients of the terms of the equation. - Determine the values of a, b, and c; a is the coefficient of x, b is the coefficient of x, and c is the constant. - Write the equation in standard form, ax 2 + bx + c = 0. - Substitute these values of a, b and c into the quadratic formula.

3.- Write in order the steps for solving a biquadratic equation - Solve the equation with the new unknown factor (apply the quadratic formula) - Call x 2 = z , which means that x 4 = z 2 - Simplify the equation and write it in standard form, ax 4 + bx 2 + c = 0 - Use these values of z to work out the values of x: z = x 2 , from which: x = the square root of z

4.-Write in order the steps for solving an equation containing algebraic fractions - Find the Lowest Common Multiple of the entire equation. - Multiply both sides of the equation by the LCM. - Check for extraneous solutions. Find any excluded values(values of the variable that would make the denominator 0) - Solve the remaining equation.

5.- Write in order the steps for solving radical equations - If you still have a radical sign left, repeat steps 1 and 2. - Check for extraneous solutions. - Get rid of your radical sign (the inverse operation to a radical or a root is to raise it to an exponent) - Isolate one of the radicals - Solve the remaining equation.

Bilingual Program MATHS 6.- The length of a rectangle is 2 times its width. The area of the rectangle is 72 square inches. Find the dimensions of the rectangle. 7.- The length of a rectangular garden is 4 yards more than its width. The area of the garden is 60 square yards. Find the dimensions of the garden. 8.- The product of two consecutive integers is 56. Find the integers. 9.- The product of two consecutive odd integers is 99. Find the integers. 10.- The product of two consecutive odd integers is 77 more than twice the larger. Find the integers. 11.- Find three consecutive integers such that three times the sum of all three equals the product of the larger two. 12.- The medium side of a right triangle is 7 more than the shortest side. The longest side is 7 less than 3 times the shortest side. Find the length of the shortest side of the triangle. 13.- One leg of a right triangle is one inch shorter than the other leg. If the hypotenuse is 5 inches, find the length of the shorter leg. 14.- The number 365, the number of days in the year, is a very curious number. It is the only number that is equal to the sum of the squares of three consecutive natural numbers and that also is equal to the sum of the squares of the following two numbers. Can you find them? 15.- Asked his age, Disraeli answered: Subtract five hundred from the product of the number of the age that I had 5 years ago by the age I will have in 5 years and the result will be four times the age I have now. Find Disraeli´s age. 16.- In Queen Victoria´s palace, there is a rectangular garden that is 50 m long by 34m wide and surrounded by a walkway. The width of the walkway is uniform. Find the width of the walkway since you know its area is 540 m 2. 17.- A piece of a rectangle of zinc is 4 cm longer than its width. With this piece, a box measuring 840 cm³ was made by cutting a square of 6 cm in every corner and then folding the edges. Find the dimensions of the box. 18.-In 11 years Pedro will be half of the square of the age that he will be in 13 years. Calculate Pedro’s age. 19.- The sides of a right triangle have measurements in centimeters of 3 even consecutive numbers. Find the values of each side.

There are some problems where you need to use the Pythagorean Theorem which states that in a right triangle: The sum of the squares of the legs is equal to the square of the hypotenuse. (Leg 1) 2 + (Leg 2) 2 = (Hypotenuse) 2

Bilingual Program MATHS UNIT 5 – SYSTEMS OF EQUATIONS AND SYSTEMS OF INEQUALITIES  SYSTEMS OF EQUATIONS Two equations for which you need a common solution are called a pair of simultaneous equations or system of equations.

Solving a pair of simultaneous equations: There is more than one way to solve any pair of simultaneous equations. English speaking people choose one of the following methods to solve them:

o the graphical method: to solve them approximately o an algebraic method (by elimination or by substitution): to solve them exactly. Substitution method for solving systems of equations: 

You substitute (or replace) one of the unknowns with an equivalent expression or value.

1.- Choose one equation and isolate one unknown. 2.- Substitute the solution from step 1 into the other equation and solve for the variable in the equation. 3.- Using the value found in step 2, substitute it into the first expression and solve for the second unknown.

Elimination method (or addition method) for solving systems of equations: 

The original equations are combined to eliminate one of the unknowns making an equation that is easier to solve.

1.- Multiply every equation by the properly number in order to get one of the unknowns with the same coefficient in both equations. Remember, you have to multiply both sides of the equation by the same

number. 2.- Add the two equations and solve the equation that you get. 3.- Substitute the found unknown by the value that you have got in one of the equations and solve for the other unknown.

Bilingual Program MATHS Graphical method for solving systems of equations: The graphical solution of linear simultaneous equations is the point of intersection found by drawing the two linear equations on the same axes.

Let’s look at an example using the graphical method:

Solve graphically:

4x - 6y = 12 2x + 2y = 6 To solve a system of equations graphically, graph both equations and see where they intersect. The intersection point is the solution. Solve each equation for "y =" and graph the lines: 4x - 6y = 12 2x + 2y = 6

2 x2 3 y  x  3

y

The point of intersection of the two lines, (3,0), is the solution to the system of equations. This means that (3,0), when substituted into either equation, will make them both true. NON LINEAR SYSTEMS OF EQUATIONS An equation in which one or more terms have a variable of degree 2 or higher is called a non linear equation. A nonlinear system of equations contains at least one nonlinear equation. When solving a system in which one equation is linear, it is easiest to use the substitution method. Solve the linear equation for either x or y, then substitute the resulting expression into the nonlinear equation.

Bilingual Program MATHS  SYSTEMS OF INEQUALITIES  SYSTEMS OF LINEAR INEQUALITIES IN ONE VARIABLE:

Solve each inequality separately, and the resultant set is the solution sets for both inequalities. For example:

 2x  3  1   x  2  1

 2 x  1  3  2 x  2  x  1   x  1  2   x  3

The solution is: x  [−1, 3]

 SYSTEMS OF LINEAR INEQUALITIES IN TWO VARIABLES:

A system of linear inequalities consists of a set of two or more linear inequalities with the same variables. The inequalities define the conditions that are to be considered simultaneously. To solve it, you only have to graph both inequalities on the same axes. The intersection of the solutions is the solution of the system.

For example:

y  x  2   y  2x  2

PROBLEM-SOLVING STRATEGY 1. Read the problem several times and analyse the facts. Occasionally, a sketch chart, or diagram will help you visualize the facts of the problem. 2. Pick different variables to represent two unknown quantities. Form two equations in two variables. 3. Solve the system using the most convenient method. 4. State the conclusion. 5. Check the solution in the words of the problem.

Bilingual Program MATHS

WORD PROBLEMS WITH SYSTEMS OF EQUATIONS 1) The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended? 2) A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totalled $487. The second order was for 6 bushes and 2 trees, and totalled $232. The bills do not list the per-item price. What were the costs of one bush and of one tree? 3) An exam worth 145 points contains 50 questions. Some of the questions are worth two points and some are worth five points. How many two point questions are on the test? How many five point questions are on the test? 4) Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold? 5) 110m of fence have been used to enclose a rectangular piece of land of 750 m2 Calculate the dimensions of the land. 6) The hypotenuse of a right triangle measures 10 cm and the sum of the catheti is 14 cm. Find the values of the catheti. 7) A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and multiple choice questions worth 11 points each. How many multiple choice questions are on the test? 8) The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number 9) The perimeter of a rectangle is 22 feet, and the area is 24 square feet. Find the length and width.

Bilingual Program MATHS UNIT 6 – CONGRUENT AND SIMILAR SHAPES  Congruent figures have the same size and the same shape. Two shapes are congruent if:  They have the same shape.  All the corresponding angles and lengths are equal.  Similar figures have the same shape, but not necessarily the same size. Two shapes are similar if:  They have the same shape.  All the corresponding angles are equal.  All the corresponding lengths are in the same ratio. Two triangles are similar if any of the following are true:   

All the corresponding angles are equal. The ratios of two pairs of sides are equal and the angles between these sides are equal. The ratios of the three pairs of sides are equal.

 Thales' theorem or Intercept Theorem: If two intersecting lines are cut by parallel lines, the line segments cut by the parallel lines from one of the lines are proportional to the corresponding line segments cut by them from the other line:

 The Altitude-0n-Hypotenuse Theorem In a right triangle, the altitude that’s perpendicular to the hypotenuse has a special property: it creates two smaller right triangles that are both similar to the original right triangle. As consequence, if an altitude is drawn to the hypotenuse of a right triangle as shown in the figure, then:

Bilingual Program MATHS UNIT 7 – TRIGONOMETRY If we have an angle θin a right triangle, we name the sides of the triangle in this way:  "Opposite" is the cathetus opposite to the angle θ  "Adjacent" is the cathetus next to the angle θ  "Hypotenuse" is the side opposite the right angle

Trigonometric ratios: sine, cosine, tangent, secant, cosecant and cotangent  Sine, Cosine and Tangent are the three main functions in trigonometry. They are often shortened to sin, cos and tan. For a triangle with an angle θ, the functions are calculated this way:

Sine:

sin(θ) = Opposite / Hypotenuse

Cosine:

cos(θ) = Adjacent / Hypotenuse

Tangent:



tan(θ) = Opposite / Adjacent

secant, cosecant and cotangent Secant: Cosecant: Cotangent:

sec(θ) = Hypotenuse / Adjacent

(=1/cosθ)

cosec(θ) = Hypotenuse / Opposite

(=1/sinθ)

cot(θ) = Adjacent / Opposite

(=1/tanθ)

Example: What is the sine of 35°? Using this triangle (lengths are only to one decimal place): sin(35°) = Opposite / Hypotenuse = 2.8 / 4.9 = 0.57... Exercise: Calculate the other 5 trigonometric ratios of 35º Pythagorean trigonometric identity The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:

sin 2   cos2   1

Bilingual Program MATHS Trigonometry Word Problems: Trigonometry word problems use the relationship between different angles and sides of right angled triangles. There are a wide variety of trigonometry word problems. The most common of them is the height and distance word problems. Question 1: From the top of a light house 60 meters high with its base at the sea level, the angle of depression* of a boat is 15 degrees. What is the distance of boat from foot of the light house? Question 2:

the

A 10 meter long ladder rests against a vertical wall so that the distance between the foot of the ladder and the wall is 2 meter. Find the angle the ladder makes with the wall and height above the ground at which the upper end of the ladder touches the wall.

Question 3: The angle of elevation* of the top of an incomplete vertical pillar at a horizontal distance of 100 m from its base is 45 degrees. If the angle of elevation of the top of the complete pillar at the same point is to be 60 degrees, then the height of the incomplete pillar is to be increased by how much? Question 4: A tree 50 feet in height casts a shadow of length 60 feet. What is the of elevation from the end of the shadow to the top of the tree with respect to the ground?

angle

Question 5: John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33ยบ. How tall is the tree? Question 6: A building is 50 feet high. At a distance away from the building, an observer notices that the angle of elevation to the top of the building is 41ยบ. How far is the observer from the base of the building? Question 7: An airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the airport? =========================================================== * Angles of depression and elevation:

Bilingual Program MATHS

Angles We usually use Greek letters such as alpha (α), beta (β), gamma (γ), and theta (θ) to represent angles. Several different units of angle measure are widely used, including degrees, radians, and grads: 1 full circumference = 360 degrees = 2

radians = 400 grads.

The following table shows the conversions for some common angles:

Degrees

30°

60°

120°

150°

210°

240°

300°

330°

Radians

Grads

33⅓ grad 66⅔ grad 133⅓ grad 166⅔ grad 233⅓ grad 266⅔ grad 333⅓ grad 366⅔ grad

Degrees

45°

90°

135°

180°

225°

270°

315°

360°

50 grad

100 grad

150 grad

200 grad

250 grad

300 grad

350 grad

400 grad

Radians

Grads

Radians - an Alternative Measure for Angle In science and engineering, radians are much more convenient (and common) than degrees. A radian is defined as the angle between 2 radii of a circle where the arc between them has length of one radius. One radian is about 57.3º There are 2 Therefore, 2

radians in a circumference. radians = 360º

Trigonometric Ratios Table

Bilingual Program MATHS

UNIT 8: FUNCTIONS

1.-Formal Definition of a Function A function relates each element of a set with exactly one element of another set (possibly the same set).

Vertical Line Test On a graph, the idea of single valued means that no vertical line would ever cross more than one value. If it crosses more than once it is still a valid curve, but it would not be a function.

2.-Domain and Range The domain of a function is the set of "input" for which the function is defined.

For a function f defined by an expression with variable x, the domain of f is the set of all real numbers variable x can take such that the expression defining the function is real. The range of f is the set of all values that the function takes when x takes values in the domain.

3.-Continuity A continuous function is a function for which, intuitively, "small" changes in the independent variable, or point of the domain, produces only a small change in the value of the function. Otherwise, a function is said to be a "discontinuous function"

Classification of discontinuities: Example 1:

Then, the point

is a removable discontinuity.

Bilingual Program MATHS Example 2:

is a jump discontinuity.

Then, the point

Example 3:

f ( x) 

x 1 x 1

Then, the point X0=1 is an essential discontinuity (sometimes called infinite discontinuity).

4.- Periodicity A periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.

A function f is said to be periodic with period P (P being a nonzero constant) if we have 5.- Increasing and decreasing functions 

A function is "increasing" if the y-value increases as the x-value increases. For a function y=f(x): when x1 < x2 then f(x1) < f(x2)

Increasing

That has to be true for any x1, x2, not just some nice ones you choose.



A function is "decreasing" if the y-value decreases as the x-value increases: when x1 < x2 then f(x1) > f(x2) Decreasing

Increasing function

Decreasing function

Bilingual Program MATHS 6.- Maximum and minimum points The graph of a function y = f(x) has a local maximum at the point where the graph changes from increasing to decreasing. The graph has a local minimum at the point where the graph changes from decreasing to increasing.

7.- Symmetry There are two types of symmetry for functions: y-axis symmetry and origin symmetry. 

Y-axis Symmetry

We say that a function has “y-axis symmetry”, or is “symmetric about the y-axis”, when its graph would look the same if it were reflected about the y-axis. So, a graph is symmetric about the y-axis if whenever the point (x,y) is on the graph, so is the point (-x,y). If f(a) = f(-a),

f(x) is an even function.

For example, our standard parabola y=f(x)=x2 has y-axis symmetry. 

Origin Symmetry

A function is said to have “origin symmetry”, or is “symmetric around the origin”, when its graph would look the same if it were rotated 180 degrees around the origin. So, a graph is symmetric about the origin if whenever the point (x,y) is on the graph, so is the point (-x,-y).

If f(-a) = - f(a), f(x) is an odd function Here are a couple of examples of graphs that symmetric about the origin:

are

8.- x- and y-Intercepts  

an x-intercept is a point in the equation where the y-value is zero a y-intercept is a point in the equation where the x-value is zero.

9.- Asymptotes An asymptote is a straight line that the graph of a function approaches but never reaches. There are two main types of asymptotes: Horizontal (y = k) and Vertical (x = a)

Bilingual Program MATHS EXERCISES: Analyze the following functions. Write their domain, range, x- and y- intercepts, increasing and decreasing intervals as maximum and minimum points. Study their continuity, symmetry, asymptotes and periodicity.

Bilingual Program MATHS UNIT 9: POLYNOMIAL, RATIONAL ,EXPONENTIAL AND LOGARITHMIC FUNCTIONS

1.-POLINOMIAL FUNCTIONS: Their domain is always all the real numbers.  Linear Functions A function that can be graphically represented in the Cartesian coordinate plane by a straight line is called a Linear function. A very common way to express a linear function is:

(m and b are constants)

The slope (also called gradient) of a straight line shows how steep a straight line is. When the gradient is 0, it is a Constant Function.  

Its formula is f(x) = c. Its graph is a horizontal line.

To calculate the Gradient:

The gradient is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run.

(x1, y1) and (x2, y2) represent two points in the straight line. It is important to keep the x-and y-coordinates in the same order in both the numerator and the denominator otherwise you will get the wrong slope. The slope m tells us if the function is increasing, decreasing or constant:



m<0

decreasing

m=0

constant

m>0

increasing

 Quadratic Functions

Bilingual Program MATHS A quadratic function, is a polynomial function of the form:

The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis.

Characteristics or Properties of Graphs of Quadratic Functions: 1.- The vertex of the graph of a quadratic function f(x) = ax2+ bx + c is the point

  b   b  V   , f    2a  2a   2.- The range of a quadratic function is:  

all real numbers greater than or equal to the y-value, if the vertex is a minimum. all real numbers less than or equal to the y-value, if the vertex is a maximum.

3.- They have an axis of symmetry. It is always a vertical line of the form x = n, where n is a real number 4.- To find the x-intercepts, solve the quadratic equation ax2+ bx + c = 0. To find the y-intercept of the parabola, find f(0)

2.-FUNCTIONS WHOSE GRAPH IS A HYPERBOLA

 Inversely proportional function: The formula is

f ( x) 

k x

When k = 1, it is called Reciprocal function:

f ( x) 

1 x

The graph of the reciprocal function is:

Bilingual Program MATHS Characteristics of the inversely proportional function: 1.- Its graph is a hyperbola. 2.- It is an odd function. 3.- Domain is the Real Numbers, except 0, because 1/0 is undefined:

Dom = {x  R /x  0} = R - {0} 4.- Its Range is also the Real Numbers, except 0.

Img = R - {0}

5.- The graph has a horizontal asymptote at the x –axis: y = 0 (This means the graph gets closer to the x-axis as the value of x increases, but it never meets the x- axis) 6.- The graph has a vertical asymptote at the y –axis: x = 0 (This means the graph gets closer to the y-axis as x gets closer to 0 but it never meets the y-axis)

 Rational function: A rational function is defined as the quotient of two polynomial functions:

f(x) 

P(x) Q(x)

If P(x) and Q(x) are first-degree polynomials, the graph is a hyperbola. For example:

y

2 4 x -3

3.- EXPONENTIAL FUNCTION Its formula is

f(x)  a x , a  1 , a  0

The function value will be positive because a positive base raised to any power is positive. To graph exponential functions, remember that unless they are transformed, the graph will always pass through (0, 1) and will approach, but not touch or cross the x-axis

Bilingual Program MATHS Characteristics of the Exponential Function:     

The domain is R The range is the set of strictly positive real numbers: (0,   ) The function is continuous in its domain The function is increasing if a > 1 and decreasing if 0<a<1 The x-axis is a horizontal asymptote.

4.-LOGARITHMIC FUNCTION The logarithmic function is the function:

f ( x)  log a x, a  1 , a  0 , and x > 0 (The function is read "log base a of x") Since x > 0, the graph of the above function will be in quadrants I and IV.

Logarithmic functions are the inverse of exponential functions. Their graphs are symmetric with respect to the angle bisector of the first quadrant.

Bilingual Program MATHS Characteristics of the logarithmic Function:     

The The The The The

range is R domain is the set of strictly positive real numbers function is continuous in its domain function is increasing if a > 1 and decreasing if 0 < a < 1 negative y-axis is a vertical asymptote

Definition of logarithm:

log a x  y  a y  x

We read this in this way: "log base a of x is y”.

The two most used logarithms are called: common logarithms (base 10)

natural logarithms (base e)

PROPERTIES OF LOGARITHMS: 1) loga 1 =0 because 2) loga a =1

because

3) loga ax =x because 4) loga MN = loga M + loga N

Think: Multiply two numbers with the same base, add the exponents.

5) loga M/N =logaM – logaN

Think: Divide two numbers with the same base, subtract the exponents.

6) loga Mp =p·logaM

Bilingual Program MATHS

UNIT 10– STATISTICS

BASIC CONCEPTS AND DEFINITIONS:     

population1 sample2 experimental unit or sampling unit3 size4 mid-interval value5

Different types of data:  Qualitative data: It is described using words.  Quantitative data: It consists of numbers.  Discrete data: It can only take particular values.  Continuous data: It can take any value.  

collecting data /ei/- Recopilar datos handling data - Recuento/manejo de datos

   

absolute frequency relative frequency absolute cumulative frequency relative cumulative frequency

A population is any entire collection of people, animals, plants or things from which we may collect data. It is the entire group we are interested in, which we wish to describe or draw conclusions about. 2

A sample is a group of units selected from a larger group (the population). By studying the sample it is hoped to draw valid conclusions about the larger group. 3

A unit is a person, animal, plant or thing which is actually studied by a researcher; the basic objects upon which the study or experiment is carried out. 4

Number of elements/items of the population/sample.

The mid-interval value is the value halfway along the interval.

Bilingual Program MATHS 

GRAPHS   o o o o



abscissas axis ordinate axis

bar chart frequency polygon histogram pie chart

CENTRALIZATION MEASURES o sample mean or average6 o mode 7 o median 8



MEASURES OF POSITION o quartile 9 o percentile10



MEASURES OF DISPERSION o range 11 o variance 12 o standard deviation 13 o average deviation or mean deviation o coefficient of variation

Useful links: http://www.stats.gla.ac.uk/steps/glossary/index.html http://www.animatedsoftware.com/statglos/statglos.htm

The mean is the sum of the values divided by the number of values. The mode is the value that occurs most often. 8 The median is the middle value or the mean of the two middle values when the values are put in order of size. 9 A quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents 1/4th of the sampled population. 10 A percentile is the value of a variable below which a certain percent of observations fall. 11 The range is the highest value minus the lowest value. 12 The variance of a sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the mean. 13 The standard deviation is the positive square root of the variance. 7

Bilingual Program MATHS

UNIT 11 – PROBABILITY 

Experiment: an action where the result is uncertain. Ex: Tossing a coin, throwing dice, picking a ball from a bag…..



Sample Space: all the possible outcomes of an experiment.



Sample Point: just one of the possible outcomes.



Event: An event is any collection of outcomes of an experiment. Formally, any subset of the sample space is an event.



Any event which consists of a single outcome in the sample space is called an elementary or simple event. Events which consist of more than one outcome are called compound events.



Sure or certain event: an event that is certain to occur. It is E, the sample space.



Impossible event: It is the event containing no outcomes. It is denoted by Ø.  Mutually Exclusive Events: These are events that cannot occur at the same time. In other words, if there is no element that is in both A and B. The intersection of A and B is Ø.  Aces and Kings are Mutually Exclusive

Hearts and Kings are not Mutually Exclusive

Two events are called not mutually

exclusive if they have at least one outcome common between them.

Operations on events: If A and B are two events in the sample space S, then: Union - The event (A  B) occurs if A or B or both A and B occur. Intersection - The event (A ∩ B) occurs only if both A and B occur.

Complement of A

A and

( A ) - This event occurs if and only if A does not occur.

A are complementary events.

Bilingual Program MATHS Independent and dependent events: 

Independent Events: These are two or more events for which the outcome of one does not affect the other. Ex: Each toss of a fair coin is an independent event.



Dependent Events: These are events that are dependent on what occurred previously. Ex: If two cards are drawn from a deck of fifty-two cards, the likelihood of the second card being an ace is dependent on the outcome of the first four cards. After taking one card from the deck there are less cards available, so the probabilities change! Don’t forget! Replacement: When you put each card back after drawing it the chances don't change, as the events are independent. Without Replacement: The chances will change, and the events are dependent.

Probability: A probability provides description of the likely particular event.

a quantitative occurrence of a

Probability goes from 0 (impossible) to 1 (certain). It is often shown as a decimal, fraction or percentage.

If all outcomes in an experiment are equally likely, the probability of an event A is calculated using the following formula known as: LAPLACE’S RULE

p( A) 

number of favorable choices for the event A Total number of possibleoutcomes

Example: what is the probability of getting a "Head" when tossing a coin? Number of ways it can happen: 1 (Head) Total number of outcomes: 2 (Head and Tail) 1 So the probability =

= 0.5

Bilingual Program MATHS Probability Axioms: 1.

The probability is positive and less than or equal to 1.

The probability of the sure event is 1.

If A and B are mutuall y exclusive,

0  p( A)  1

p(E)  1 then: p( A  B)  p( A)  p(B)

Probability Properties 1 The probability of the complementary event is:

2 The probability of an impossible event is zero.

p( A )  1  p( A)

p(Ø.)  0

3 The probability of the union of two events is the sum of their probabilities minus the probability of their intersection.

p( A  B)  p( A)  p(B)  p( A  B) )

4 The sum of the probabilities of all possible outcomes equals 1.

5 If an event is a subset of another event, its probability is less than or equal to it.

If A  B, p( A)  p(B) 6 Two events A and B are independent

7 A tree diagram is useful for displaying all outcomes for a “multistage” experiment and determining their probabilities.

 p( A  B)  p( A)  p(B)

Bilingual Program MATHS Conditional Probability We have already defined dependent and independent events and seen how probability of one event relates to the probability of the other event. Having those concepts in mind, we can now look at conditional probability. Conditional probability is denoted by the following:

 

p AB

It is read as the probability that A occurs given that B has already occurred.

Summary of probabilities Event

Probability

not A

A or B

A and B

A given B

Note: The most common deck of fifty-two playing cards includes thirteen ranks of each of the four suits: clubs (♣), diamonds (♦), hearts (♥) and spades (♠) and, usually, two jokers. You shuffle the cards before dealing them to play.

Bilingual Program MATHS PROBABILITY PROBLEMS 1) If a coin is flipped twice, what is the probability that it will land â&#x20AC;Ś a) heads once and tails once? b) heads at least once? 2) A bag contains 4 white counters, 6 black counters, and 1 green counter. What is the probability of drawing: a) A black counter? b) A white counter or a black counter? c) A white counter or a green counter? 3) A deck of cards contains 52 cards, 13 from each suit. If a card is flipped over, what is the probability that it is a spade? That it is a face card (J, Q, K) of any suit? 4) A card is drawn from a pack of 52 cards. What is the probability of getting a queen of club or a king of heart? 5) A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. What is the probability that all of them are red? 6) Two cards are drawn together from a pack of 52 cards. What is the probability that one is a spade and one is a heart? 7) Two dice are rolled. What is the probability that the total score is a prime number? 8) A medical trial into the effectiveness of a new medication was carried out. 120 females and 90 males took part in the trial. Out of those people, 50 females and 30 males responded positively to the medication. Make a contingency table. a) What is the probability that the medicine gives a positive result? b) What is the probability that the medicine gives a positive result for females? c) What is the probability that if the medicine gives a positive result, the person is a woman? d) What is the probability that the medicine gives a negative result for males? e) What is the probability that if the medicine gives a positive result, the person is a man? 9) A car dealership is giving away a trip to Rome to one of their 120 best customers. In this group, 65 are women, 80 people are married and 45 are married women. a) What is the probability that the winner is a single person? b) What is the probability that the trip will be awarded to a bachelor (single man)? c) If the winner is married, what is the probability that it is a woman? d) What is the probability that the winner is a married woman? e) Knowing the winner is a woman, what is the probability that it is married?

Bilingual Program MATHS PROBABILITY

CHOOSE THE RIGHT ANSWER:

1) A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

2) What is the probability of getting a sum 9 from two throws of a dice?

3) In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

4) Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

5) Three unbiased coins are tossed. What is the probability of getting at most two heads? 3 1 3 7

4 4 8 8 6) Two dice are rolled simultaneously. What is the probability of getting two numbers whose product is even?

1 2

3 4

3 8

5 16

7) In a class, there are 5 boys and 7 girls. Three students are selected at random. The probability of selecting 1 girl and 2 boys is:

14 132

;

7 22

;

3 12

;

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8) In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability of getting a prize?

9) From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the cards being kings?

1 15

25 57

35 256

1 221

Bilingual Program MATHS MORE EXERCISES OF PROBABILITY 1.-If a single 6-sided die is rolled, what is the probability of rolling a number that is not 8? A) 5/6

B) 1

C) 0

D) None of the above

2.- A single letter is chosen at random from the word TEACHER. All of the following are mutually exclusive events except: A) Choosing a T or choosing a consonant

B) Choosing a T or choosing a vowel

C) Choosing an E or choosing a C

D) None of the above

3.-The probability of a New York teenager owning a skateboard is 0.37, of owning a bicycle is 0.81, and of owning both is 0.36. If a New York teenager is chosen at random, what is the probability that he/she owns a skateboard or a bicycle? A) 1.18

B) 0.7

C) 0.82

D) None of the above

4.- A single 6-sided die is rolled. What is the probability of getting a number greater than 3 or an even number? A) 1

B) 2/3

C) 5/6

D) None of the above

5.- Four cards are chosen from a standard deck of 52 playing cards with replacement. What is the probability of getting 4 hearts? A) 13/52

B) 1/16

C) 1/256

D) None of the above

6.- Two cards are chosen at random from a deck of 52 cards without replacement. What is the probability of getting two kings? A) 4/663

B) 1/221

C) 1/69

D) None of the above

7.- Spin a spinner numbered 1 to 7, and toss a coin. What is the probability of getting an odd number on the spinner and a tail on the coin? A) 3/14

B) 2/7

C) 5/14

D) None of the above

8.- A school survey found that 7 out of 30 students walk to school. If four students are selected at random without replacement, what is the probability that all four walk to school? A) 343/93960

B) 1/783

C) 7/6750

D) None of the above

9.- At a middle school, 18% of all students play football and basketball, and 32% of all students play football. What is the probability that a student who plays football also plays basketball? A) 56%

B) 178%

C) 50%

D) None of the above

Bilingual Program MATHS

Does the order affect? YES

Do all the elements enter?

YES

Can they be repeated?

YES

Permutations with repetition

Permutations without repetition

Variations with repetition

Variations without repetition

Combinations with repetition

Combinations without repetition

Pn=n(n-1)(n-2)...3Âˇ2Âˇ1=n!

VRm,n=mn

Vm,n=m(m-1)(m-2)...(m-n+1)

Bilingual Program MATHS COMBINATORICS PROBLEMS 1) How many 3- digit numbers can we write using some of the figures 1, 2, 3, 4 without repeating any one? 2) In how many different ways can the letters in the Word ENGLISH be arranged? 3) In how many ways can the 4 seats on the first row be filled by 4 students chosen of a group of 12 students? 4) Out of 5 mathematicians and 7 engineers, a committee consisting of 2 mathematicians and 3 engineers is to be formed. In how many ways can this be done? 5) How many possible positions do we have in a championship in which 4 teams take part? 6) Twelve people apply for a job where only 3 people are needed. How many ways are there to assign three jobs to twelve employees? Note: Each employee cannot be given more than one job! 7) How many different combinations of management can there be to fill the positions of president, vice-president and treasurer of a football club knowing that there are 12 eligible candidates? 8) How many five-digit numbers can be formed with the digits 1, 2 and 3? How many of those numbers are even? 9) How many numbers greater than 4000 can be formed with the digits 3, 4, 6, 8, 9, if a digit cannot occur more than once in a number? 10) How many different number-plates for cars can be made if each number-plate contains four of the digits 0 to 9 followed by a letter A to Z, assuming that repetition of digits is allowed? 11) How many different sets of 4 letters can be selected from the alphabet, supposing the alphabet has 27 letters? 12) In how many different ways can a supermarket manager display 5 brands of cereals in 3 spaces on a shelf? 13) How many numbers greater than 1000 can be formed with the digits 3, 4, 6, 8, 9, if a digit cannot occur more than once in a number? http://www.coolmath.com/algebra/20-combinatorics