clil math - liceoscacchibari.edu.it › wp-content › uploads › 2014 › 11 › clil-math.pdfclil...

CLILMath

Volume 1Logic, p. 1Functions, p. 2Powers, p. 3Powers, p. 5Fractions, p. 7Real numbers, p. 11Algebraic expressions, p. 14Equations and identities, p. 17System of equations, p. 19Linear inequalities, p. 20Absolute value equations, p. 21Data organization, p. 23

Volume 2Surds, p. 25

Apertura.qxd 5-04-2011 14:34 Pagina 1

Math literatureLogicA statement is a sentence which can be decided definitely as true or false.

A conjunction p ! q (it is read “p and q”) is a true statement onlyif p and q are both true, otherwise it is false. Look at the truth value table.The intersection used in set theory is defined in terms of a logi-cal conjunction: x "A # B if and only if (x "A) ! (x "B).Logical conjunction satisfies the associative, commutative, idem-potence and distributive properties, as well as De Morgan’s laws.

A disjunction p $ q (it is read “p or q”) is a false statement onlyif p and q are both false, otherwise it is true.Look at the truth value table.The union used in set theory is defined in terms of a logicaldisjunction: x "A % B if and only if (x "A) $ (x "B).Logical disjunction satisfies the associative, commutative anddistributive properties, as well as De Morgan’s laws.

A negative ¬ p is a true statement if p is false, and is a false sta-tement if p is true.Look at the truth value table.

An implication p & q is a false statement only if p is true and qis false, otherwise it is true.Look at the truth value table.

An equivalence p ' q is a true statement only if both the state-ments p and q have the same truth value (either both true or bothfalse).Look at the truth value table.

1

LogicCLILMath

p q p!qT T TT F FF T FF F F

p q p$qT T TT F TF T TF F F

p q p&qT F FT T TF T TF F T

p q p'qT F FT T TF T FF F T

p ¬ pT FF T


2

FunctionsCLILMath

Math literatureFunctionsA variable y is a function of the variable x, defined in a set of numbers A, if there is a rulewhich assigns to each value of x in the set A an unique value of y.The variable y, whose value depends upon the chosen value of x in the set A, is called thedependent variable. x is called the independent variable.The set of values which the independent variable x can assume is called the domain of thefunction. The set of values which the dependent variable y can assume is called the range of thefunction.If y is the unique value associated with x under function f, you can write y = f(x) and youwill read it as “f of x”. y = f(x) is called the value of f at x.Let’s look at the following examples.

• Given f(x) = 3x2 + 1 find f((2), f((1), f(0), f(1).f((2) = 3((2)2 + 1 = 13f((1) = 3((1)2 + 1 = 4f(0) = 3(0)2 + 1 = 1f(1) = 3(1)2 + 1 = 4

example


3

PowersCLILMath

Math literatureExponents of natural numbersAn corresponds to raising the number A to the power n and describes the result of repeatedlymultiplying the number A by itself:

An = A ) A ) A... ) A(A appears n times)

The number A that is successively multiplied by itself is called the base.n is called the exponent and indicates the number of times the base is to be multiplied byitself.Let’s look at the following examples.

• 2 ) 2 ) 2 ) 2 ) 2 ) 2 ) 2 ) 2 = 28

• 24 = 2 ) 2 ) 2 ) 2 = 16

example

These laws can be used to simplify expressions which contain powers.

• Any number, except 0, raised to the zero power is equal to 1:

A0 = 1, **A ++ 0

• Any number raised to the one power is equal to same number:

A1 = A, **A

• 0 raised to the power of any natural number, except 0, is equal to 0:

0n = 0, **n "N0 ++ 0

• You can multiply powers with the same base by adding the exponents:

Am )) An = Am+n

• You can multiply powers with the same exponent by multiplying the bases:

Am )) Bm = (A )) B)m

• You can divide powers with the same base by subtracting the exponents:

Am : An = Am((n

• You can divide powers with the same exponent by dividing the bases:

Am : Bm = (A : B)m

• You can simplify the power of a power by multiplying the exponents:

(Am)n = Am )) n


4

PowersCLILMath

Let’s look at the following examples.

• Multiplying powers with the same base:24 ) 25 = 24+5 = 29

• Dividing powers with the same base:45 : 43 = 45(3 = 42

211 : 24 = 211(4 = 27

• Multiplying powers with the same exponent:24 ) 34 = (2 ) 3)4 = 64

25 ) 35 = [2 ) 3]5 = 65

• Power of a power:(24)3 = 24·3 = 212

example


5

PowersCLILMath

Math literatureExponents of relative numbersAn corresponds to raising the number A to the power n and describes the result of repeate-dly multiplying the number A by itself:


The number A that is successively multiplied by itself is called the base.n is called the exponent and indicates the number of times the base is to be multiplied byitself.Let’s look at the following examples.

• ((3) ) ((3) ) ((3) ) ((3) ) ((3) = ((3)5

• (+2) ) (+2) ) (+2) ) (+2) ) (+2) ) (+2) ) (+2) ) (+2) = (+2)8

• 24 = 2 ) 2 ) 2 ) 2 = 16• ((2)5 = ((2) ) ((2) ) ((2) ) ((2) ) ((2)

example

These laws can be used to simplify expressions which contain powers.• Any number, except 0, raised to the zero power is equal to 1:

A0 = 1, **A ++ 0• Any number raised to the one power is equal to same number:

A1 = A, **A• 0 raised to the power of any natural number, except 0, is equal to 0:

0n = 0, **n "N0 ++ 0• If you are working with powers, you must watch for the sign of the base (if it is different

from 0): a) if the sign of the base is negative and the exponent is even, the result is positive; b) if the sign of the base is negative and the exponent is odd, the result is negative;c) if the sign of the base is positive and the exponent is even, the result is positive; d) if the sign of the base is positive and the exponent is odd, the result is positive.

• You can multiply powers with the same base by adding the exponent:Am )) An = Am+n

• You can multiply powers with the same exponent by multiplying the bases:Am )) Bm = (A )) B)m

• You can divide powers with the same base by subtracting the exponents:Am : An = Am((n


6

PowersCLILMath

• You can divide powers with the same exponent by dividing the bases:Am : Bm = (A : B)m

• You can simplify the power of a power by multiplying the exponents:(Am)n = Am )) n


• ((2)5 = ((2) ) ((2) ) ((2) ) ((2) ) ((2) = (32• Multiplying powers with the same exponent

24 ) 34 = (2 ) 3)4 = 64

((2)5 ) (+3)5 = [((2)(+3)]5 = ((6)5

• Dividing powers with the same base45:43 = 45(3 = 42

((2)11 : ((2)4 = ((2)11(4 = ((2)7

• Multiplying powers with the same base((2)4 ) ((2)5 = ((2)4+5 = ((2)9

• Power of a power(24)3 = 24·3 = 212

[((2)3]2 = ((2)3·2 = ((2)6 = + 64

example


7

FractionsCLILMath

Math literatureEquivalent fractionsEquivalent fractions are fractions which are of equal value.

When two fractions e are equivalent you can write and a ) d = b ) c.

Simplify a fractionTo simplify or reduce a fraction is to get the numerator and denominator as small as pos-sible: that is possible if you can divide both of them by the same number.If you divide numerator and denominator by the Greatest Common Divisor then the frac-tion gets irreducible.

Least Common DenominatorTo reduce more fractions to the Least Common Denominator (LCD) do the following:• factor each denominator into prime numbers;• multiply the largest of each prime factor that appears in the factorizations: the least com-

mon denominator is the final result.

For each starting fraction do the following:• divide the least common denominator by the denominator and write down the result;• multiply each of the numerator and denominator by the result.

Let’s look at the following example.

ab

cd

=cd

ab

• Reduce to the Least Common Denominator and .

Factor each denominator into prime numbers: 450 = 2 ) 32 ) 52; 250 = 2 ) 53.Multiply the largest of each prime factor that appears in the factorizations:2 ) 32 ) 53 = 2250.2250 is the least common denominator.Divide the least common denominator by each of the starting denominators and writedown the results:2250 : 450 = (2 ) 32 ) 53) : (2 ) 32 ) 52) = 5; 2250 : 250 = (2 ) 32 ) 53) : (2 ) 53) = 32 = 9

Multiply each of the starting numerators and denominators by its result:

and are reduced to the Least Common Denominator.9

2250365

2250

73 5450 5

3652250

1 9250 9

92250

))

= ))

=;

1250

73450

example


8

FractionsCLILMath

Comparing fractions

e are two fractions.

To decide which fraction is greater, do the following:• find the Least Common Multiple (LCM) of both denominators;• rewrite the fractions as equivalent fractions with the LCM as the denominator;• compare the last (new) numerators.

The greater fraction is the one which has the greater numerator.


cd

ab

and are equivalent fractions.

and are equivalent fractions.9

22501

250

3652250

73450

example

• Which of the following is the greater fraction?

and

Find the Least Common Multiple (LCM) of both denominators: 63.

Rewrite the fractions as equivalent fractions with the LCM as the denominator:

and .

Compare the last (new) numerators: 45 < 49.

As you can see: 45 < 49 so: .

The greater fraction is the one which has the greater numerator: .57

79

<

4563

4963

<

79

4963

=57

4563

=

79

57

example

Exponents of fractionsAn corresponds to raising the number A to the power n and describes the result of repeate-dly multiplying the number A by itself:



9

FractionsCLILMath

The number A that is successively multiplied by itself is called the base.n is called the exponent and indicates the number of times the base is to be multiplied byitself.

When the base is a fraction, remember that: .


AB

AB

n n

n

,-.

/01

=

• ((3) ) ((3) ) ((3) ) ((3) ) ((3) = ((3)5

(+2) ) (+2) ) (+2) ) (+2) ) (+2) ) (+2) ) (+2) ) (+2) = (+2)8

24 = 2 ) 2 ) 2 ) 2 = 16((2)5 = ((2) ) ((2) ) ((2) ) ((2) ) ((2)

((1,3)2 = ((1,3) ) ((1,3)

(,-.

/01

= (,-.

/01) (,-.

/01) (,-.

/01

23

23

23

23

3

example

These laws can be used to simplify expressions which contain powers.• Any number, except 0, raised to the zero power is equal to 1:

A0 = 1, **A ++ 0• Any number raised to the one power is equal to same number:

A1 = A, **A• 0 raised to the power of any natural number, except 0, is equal to 0:

0n = 0, **n "N0

• If you are working with powers, you must watch for the sign of the base (if it is differentfrom 0):a) if the sign of the base is negative and the exponent is even, the result is positive; b) if the sign of the base is negative and the exponent is odd, the result is negative;c) if the sign of the base is positive and the exponent is even, the result is positive; d) if the sign of the base is positive and the exponent is odd, the result is positive.

• You can multiply powers with the same base by adding the exponent:Am )) An = Am+n

• You can multiply powers with the same exponent by multiplying the bases:Am )) Bm = (A )) B)m

• You can divide powers with the same base by subtracting the exponents:Am : An = Am((n

• You can divide powers with the same exponent by dividing the bases:Am : Bm = (A : B)m

• You can simplify the power of a power by multiplying the exponents:(Am)n = Am )) n


10

FractionsCLILMath

• A negative power is the reciprocal of the corresponding positive power:


AA

A AB

BA

nn n n

((

=,-.

/01

* +,-.

/01

=,-.

/01

1 0, , , ** + ! +A B0 0

• ((2)5 = ((2) ) ((2) ) ((2) ) ((2) ) ((2) = (32

((1,3)2 = ((1,3) ) ((1,3) = + 1,69

Multiplying powers with the same base((2)4 ) ((2)5 = ((2)4+5 = ((2)9

Dividing powers with the same base45 : 43 = 45(3 = 42

((2)11 : ((2)4 = ((2)11(4 = ((2)7

Multiplying powers with the same exponent24 ) 34 = (2 ) 3)4 = 64

((2)5 ) (+3)5 = [((2)(+3)]5 = ((6)5

((1,3)2 ) (0,3)2 = [((1,3) ) (0,3)]2 = ((0,39)2

Power of a power(24)3 = 24·3 = 212

Negative power

(,-.

/01

= (,-.

/01

=(

25

52

254

2 2

5 15

125

22

( =,-.

/01

=

(,-.

/01

2

344

5

677

= (,-.

/01

= (,-.

/01

)23

23

23

3 2 3 2 6

(,-.

/01

) +,-.

/01

= (,-.

/01) +,-.

/01

2

3

23

57

23

57

3 3

445

67 = (

,-.

/01

3 31021

(,-.

/01

(,-.

/01

= (,-.

/01

= (,-.

(32

32

32

32

12 7 12 7

://01

5

(,-.

/01

) (,-.

/01

= (,-.

/01

= (,-.

/0

+35

35

35

35

2 4 2 4

11

6

(,-.

/01

= (,-.

/01) (,-.

/01) (,-.

/01

= (23

23

23

23

83

227

example


11

Realnumbers

CLILMath

Math literatureReal numbersA real number is rational or irrational.

Among the real numbers are:• integer numbers; fractions; limited decimal numbers; periodic unlimited decimal num-

bers; non-periodic unlimited decimal numbers; the roots of rational numbers.

The following are real numbers:• –1, 14, –110 (integer numbers)

• , (fractions)

• 0,1; –42,1 (limited decimal numbers)

• , (periodic unlimited decimal numbers)

• 3,125689…, –11,142781… (non-periodic unlimited decimal numbers)

• , , , , (roots of rational numbers)! 17

6335! 13113

0 25,0 2,

! 125

13

example

An irrational number in the shape of root is called surd.• is called surd of the nth order.

• is called radical sign.• n is called radical index or only index.

• k is called radicand.• is a surd too. h is called coefficient.

If the radical index is an even number then the radicand must be larger than or equal to zero.

Law of surds

• . This law is called invariance.

• .

If a is the Great Common Divisor of d and n, a = GCD(d, n), then is called irre-ducible.

Signs of and must be obviously equal.mn ad a ::mnd

mn ad a ::

m m R m a N a da

Znd n ad a= " # > $ " # % $ #:: m 0 0 0

m m R m a d Nnd n ad a= " # > $ " #&& m 0 0,

h kn&

n

kn


12

Realnumbers

CLILMath


A numeric interval is a subset of R.

The numerical intervals are divided into limited and non-limited and there are two differentways to indicate them.

WARNING: ! is not a number, but only a symbol that is used to indicate that an intervalof numbers is not finite.

Non-limited intervals (square brackets indicate that the extremes are included; those roundindicate that the extremes are not included):

[a; +!) is the set of numbers greater than or equal to a

(a; +!) is the set of numbers greater than a

(–!; a] is the set of numbers less than or equal to a

(–!; a) is the set of numbers less than a.

Limited intervals:

[a; b] is the set of numbers greater than or equal to aand less than or equal to b.

[a; b) is the set of numbers greater than or equal to aand less than b.

(a; b] is the set of numbers greater than a and less thanor equal to b.

16 2 2 4

8 2

18 418 29 9

3

= = =

! = !

5 5

8 2 2

44

3 33

=

= =

example

a

x ' a

a

x > a

a

x ( a

a

x < a

a b

a ( x ( b

a b

a ( x < b

a b

a < x ( b


13

Realnumbers

CLILMath

(a; b) is the set of numbers greater than a and less than b.

a, b is not a numeric interval, but the set composedonly of numbers a and b.

• is the set of real numbers less than

• is the set of real numbers less than or equal to

• is the set of real numbers greater than or equal to and less than or equal

to

• is the set of real numbers greater than or equal to and less than 12

! 14

!)*+

,-.

14

12

,

12

! 14

!)*+

/01

14

12

,

12

!234

/01!, 1

2

12

!234

,-.

!, 12

example

a b

a < x < b

a b

x = a x = b


14

Algebraicexpressions

CLILMath

Math literatureAlgebraic expressionsAn algebraic expression contains additions, subtractions, multiplications or divisionsbetween numbers and variables.If there is not a division between two variables of an algebraic expression, it is called aninteger algebraic expression.If not explicitly mentioned, algebraic expressions will be understood as integer algebraicexpressions.The parts of an integer algebraic expression separated by the + or ( signs are called terms.The number in front of a variable in a term is called the coefficient of the term.When the terms differ only in their coefficients, they are called like terms, otherwise theyare unlike terms.The set of numbers that each variable of an algebraic expression can take is called thedomain of the algebraic expression.


• 2x3 + 4x + 5y is an integer algebraic expressions in x and y. Its terms are 2x3, 4x and5y.2 is the coefficient of 2x3, 4 is the coefficient of 4x and 5 is the coefficient of 5y.

example

• 2x2 + 5 is a binomial in x.• 2x is a monomial.• 0x2 + 0x is a null polynomial.

example

A polynomial in a variable x is an expression formed by numbers and the variable x, con-tainings no other than addition, subtraction and multiplication.The highest exponent of x is called the degree of the polynomial.A polynomial in x of degree 1 is called a linear polynomial.If all coefficients of a polynomial are zero, the polynomial is called a null polynomial.A polynomial formed by one term, not null, is called a monomial.A polynomial formed by two unlike and not null terms is called a binomial.


Addition and subtractionIf the sign before a pair of brackets is +, you can remove the brackets and rewrite the polynomial.If the sign before a pair of brackets is (, you can remove the brackets if you change thesigns of the terms of the polynomial written inside the brackets.


15


CLILMath

The terms of a polynomial can be added only if they are like terms.To add or subtract several polynomials in x:• arrange each polynomial in descending powers of x;• note zero when there is no term;• remove the brackets of the second polynomial (change the signs of the terms of the poly-

nomial if the sign before the brackets is ();• write the second polynomial below the first one tabulating the same powers of x;• add or subtract the tabulated like terms.


• ((4x + 2 + 2x4 + x3) ( (1 + 2x3 ( 6x2).Arrange the first polynomial in descending powers of x (note zero when there is noterm and do not change the signs of its terms because before the brackets there is +):

2x4 + x3 + 0x2 ( 4x + 2

Arrange the second polynomial in descending powers of x (note zero when there is noterm and change the signs of its terms because before the brackets there is ():

0x4 ( 2x3 + 6x2 + 0x ( 1

Tabulate and add or subtract the same powers of x of each polynomial:

example

2x4 +x3 +0x2 (4x +2

0x4 (2x3 +6x2 +0x (12x4 (x3 +6x2 (4x +1

MultiplicationTo multiply two polynomials, you have multiply each term of the first polynomial by eachterm of the second one (review the laws of exponents).


DivisionTo divide a polynomial by a monomial having a lower degree than the degree of the polyno-mial, you have divide each term of the polynomial by the term of the monomial (review thelaws of exponents).

• ((4x4 + x3) ) (1 ( 6x2) = (4x4 ) (1 ( 6x2) + x3 ) (1 ( 6x2) = (4x4 + 24x6 + x3 ( 6x5 == 24x6 ( 6x5 ( 4x4 + x3

example


16


CLILMath

• ((4x4 + x3) : ((6x2) = (4x4 : ((6x2) + x3 : ((6x2) =23

16

2x x(example

• (6x3 ( 31x2 + 38x ( 5) : (3x ( 5)

( ) : ( )6 31 38 5 3 5 2 7 13 2 2x x x x x x( + ( ( = ( +

6 31 38 5 3 53 2x x x x( + ( (

6 31 38 5 3 53 2x x x x( + ( (

6 31 38 5 3 53 2x x x x( + ( (

6 31 38 5 3 53 2x x x x( + ( (

6 31 38 5 3 53 2x x x x( + ( (

example

Let’s look at the following example.To divide a polynomial in x by a linear binomial, after you arrange the divisor and dividendin descending powers, proceed as in the following example (review the laws of exponents).

2x2

2x2

–6x3 + 10x2

2x2–6x3 + 10x2

– 21x2 +38x – 5

2x2 – 7x + 1–6x3 + 10x2

– 21x2 +38x – 5+ 21x2 –35x

+3x – 5–3x + 5// //


17

Equationsand identities

CLILMath

Math literatureEquations and identitiesAn identity is an equality between two algebraic expressions which is true for everynumeric value given to their variables.


• 2a = 6a ( 4a and (a ( b)(a + b) = a2 ( b2 are identities because they are true for everynumeric value given to their variables.

example

An equation is an equality between two algebraic expressions which is true for somenumeric values given to their variables.The algebraic expression before the equals sign is called the left side or first side; the oneafter the equals sign is called the right side or second side.A linear equation in x is an equation in x which contains only linear expressions in x.The solution set of an equation consists of all the values that satisfy the equation; in otherwords, that when filled in for the variables, return the equation an identity.If S 8 R (the solution set of the equation is a subset of R) then the equation is called a deter-minate equation.If S = 9 (the solution set of the equation is the empty set) then the equation is called impos-sible equation.If S = R (the solution set of the equation is R) then the equation is called an indetermina-te equation.To solve an equation means to find its solution set.In the solving process of an equation:• you can add or subtract the same quantity to both sides without altering the solution set

of the equation;• you can multiply or divide both sides by the same non-zero quantity without altering the

solution set of the equation.

FormulaeA formula is any equation involving the use of variables.To make a variable the subject of the formula means to isolate it on the left side of theformula.



18

Equationsand identities

CLILMath

• Solve the equation 5a = 6a ( 4a + 1.Add the like terms on the right side: 5a = 2a + 1.Subtract the same quantity 2a from both sides: 5a ( 2a = 2a ( 2a + 1 & 3a = 1.

Divide both sides by 3: .

The solution set is .

• Make y the subject of the formula: x ( 2y = 4.Isolate y on left side (subtract x from both sides): x ( 2y ( x = 4 ( x & (2y = 4 ( x

Divide both sides by (2: .

y is now the subject of the formula.

((

= ((

& = ( +22

42

22

y x y x

S =:;<

=>?

13

33

13

a =

example


19

System ofequations

CLILMath

Math literatureSystem of equationsA system of equations is a set which consists of at least two linear equations.The solution set of a system of two equations consists of all the values that satisfy bothequations. Therefore it is the intersection of the solution sets of its equations.If S 8 R (the solution set of the system is a subset of R) then the system is called a deter-minate system.If S = 9 (the solution set of the system is the empty set) then the system is called animpossible system.If S = R (the solution set of the system is R) then the system is called an indeterminatesystem.To resolve a system of equations in x by the substitution method:• isolate x (or y) on the left side of an equation;• substitute in the other equation the extracted expression in x (or y) and solve the extracted

equation in y (or x);• substitute in the other equation the extracted value of y (or x).

The solution set solution is S = {(x; y)}.


• Solve the determinate system of equations by the substitution method:

Isolate x on the left side of an equation, for example the first:

2x ( 5x = 7y ( 10 & (3x = 7y ( 10 &

Substitute in the other equation the extracted expression in x and solve the extractedequation in y:

Substitute in the other equation the extracted value of y:

The solution set is S = {(1; 1)}

x = ( ) =10 7 13

1

410 73

5 1 40 28 15 3 43 4( ( = ( & ( ( = ( & ( = (y y y y y 33 1 & =y

x y= (10 73

2 7 5 104 5 1x y xx y( = (( (

:;<

example


20

Linearinequalities

CLILMath

Math literatureLinear inequalities> , < , ! and " are signs of inequalities.Solving a linear inequality is like solving a linear equation while remembering the follo-wing laws of inequalities:• A > B # A + C > B + C• A > B # kA > kB if k > 0• A > B # kA < kB if k < 0


• $3 $ 2x < x + 1 % $2x $ x < 1 + 3 % $3x < 4 % x > $ 43

example


21

Absolutevalue ...

CLILMath

Math literatureAbsolute value equationsAn absolute value equation is an equation where the variable, or an algebraic expression ofit, is present as the absolute value.To solve such an equation means to solve two mixed systems both formed by one equationand one inequality.In the first system:• the equation is obtained from the original by removing the absolute value bars;• the inequality is obtained by imposing greater than or equal to 0 on the polynomial inside

the absolute value bars.In the second system:• the equation is obtained from the original by removing the absolute value bars and

changing the sign of each term;• the inequality is obtained by imposing less than 0 on the polynomial inside the absolute

value bars.The solution set of the original equation is given from the union of the solution sets of thetwo systems.


• |2x $ 1| $ 7 = 0

First system:

Its solutions set is S1 = {4}.

Second system:

Its solutions set is S2 = {$3}.The solution set of original equation is S = S1 & S2 = {$3, 4}.

x

x

x

x

x<

$ + $ =

'()

*)% <

$ =

'()

*)%

12

2 1 7 0

12

2 6 <<

= $

'()

*)

123x

x

x

x

x

x!

$ $ =

'()

*)% !

=

'()

*)% !1

22 1 7 0

12

2 8

1 22

4x =

'()

*)

example

Absolute value inequalitiesAn absolute value inequality is an inequality where the variable, or an algebraic expressionof it, is present as the absolute value.To solve such an inequality means to solve two systems of inequalities both formed by twoinequalities.


22

Absolutevalue ...

CLILMath

In the first system:• the first inequality is obtained from the original one by removing the absolute value bars;• the second inequality is obtained by imposing greater than 0 on the polynomial inside the

absolute value bars.

In the second system:• the first inequality is obtained from the original one by removing the absolute value bars

and changing the sign of each term;• the second inequality is obtained by imposing less than 0 on the polynomial inside the

absolute value bars.

The solution set of the original inequality is given from the union of the solution sets of thetwo systems.


• |4x + 2|$1 < 0

First system:

Its solutions set is

Second system:

Its solutions set is

The solution set of the original inequality is

S S S= & = $ $+,-

./0

& $ $123

./0

= $ $1 212

14

34

12

34

14

, , ,1123

./0

S234

12

= $ $123

./0

,

4 2 04 2 1 0

12

4 2 1 0

xx

x

x

+ <$ + $ <

'(*

% < $

$ $ $ <

'()

*( )

))

%< $

> $

'

())

*))

x

x

1234

S112

14

= $ $+,-

./0

,

4 2 04 2 1 0

12

4 1

xx

x

x

x+ !+ $ <

'(*

% ! $

< $

'()

*)%

!! $

< $

'

())

*))

1214

x

example


23

Dataorganization

CLILMath

Math literatureData organizationStatistics is a branch of mathematics which deals with collection, organization, analysisand graphical representation of numerical data.Raw data are usually organised by dividing the numerical data into a number of groupswhich are called classes. Then the groups or classes can be represented in a frequency divi-sion table showing the frequency of each class.There are many ways to graphically represent a group of data, for example by: bar charts,histograms, pie charts.In bar charts and in histograms, all rectangles have equal width and heights are proportio-nal to the corresponding frequencies. In bar charts, rectangles can be vertical or horizontal. In histograms, rectangles can be vertical. Their width represents classes.In pie charts, each sector at the centre is proportional to the corresponding frequency.


• The following list shows the height (cm) of 28 teen-agers:190; 171; 175; 160; 161; 180; 181; 175; 176; 171; 161; 191;181; 180; 165; 175; 171; 165; 190; 161; 171; 175; 180; 175; 181; 175; 171; 186.The table on the right represents the data per class:

The following histogram shows the graphic representation ofthe data:

exampleheight f

160-164 4

165-169 2

170-174 5

175-179 7

180-184 6

185-189 1

190-194 3

0

1

2

3

4

5

6

7

8

160-164 165-169 170-174 175-179 180-184 185-189 190-194


24

Dataorganization

CLILMath

An average is the representative value of a group of data.There are three important types of averages: the mode, the arithmetic mean and themedian.The mode of a group of data is the element of the group of data which has the highest fre-quency (the element of the group of data which appears more times than other elements).The arithmetic mean of a group of data is the sum of the data divided by the number ofdata addends.

If x1, x2, …, xn are the data, then.

If the number of data, arranged in order of magnitude, is odd then the median is the element

of the group in the position (the value in the middle).

If the number of n data, arranged in order of magnitude, is even then the median is the

arithmetic mean of elements in the and positions.


n2

1+n2

n +12

mx x x

nan=

+ + +1 2 ...

• Find the mode, the arithmetic mean and the median of the following data (heights of28 teen-agers): 190; 171; 175; 160; 161; 180; 181; 175; 176; 171; 161; 191; 181; 180;165; 175; 171; 165; 190; 161; 171; 175; 180; 175; 181; 175; 171; 186.

The mode is 175.The arithmetic mean is 174 cm.The median is 175 (remember: before you can work out the median, you must arrangethe data in order of magnitude).

example


25

SurdsCLILMath

Math literatureSurdsAn irrational number in the shape of a root is called a surd.

is called surd of the nth order.

is called radical sign. n is called radical index or only index.k is called radicand.

is a surd too. h is called coefficient.If the radical index is an even number then the radicand must be larger than or equal to zero.

Law of surds

. This law is called invariance.

If a is the great common divisor of d and n, a = GCD(d, n), then is called irredu-cible.

Signs of and must be obviously equal.


mn ad a ::mnd

mn ad a ::

m m m R m a N a da

Nnd n ad a= * " > !* " + ! ":: 0 0 0

m m m R m a d Nnd n ad a= * " > !* ")) 0 0,

h kn)

n

kn

5 5

8 2 2

16 2 2

44

3 33

18 418 29

=

= =

= =

example

Quadratic surdsThe radical index n = 2 of a surd is never written.Surds having 2 as radical index are called quadratic surds. is a quadratic surd andk @ 0.Quadratic surds may be simplified when they are multiplied or divided:

,

Quadratic surds may be added, if they have the same index and the same radical, by multi-

AB

AB

=A B A B) = )

k


26

SurdsCLILMath

plying the sum of their coefficients by the common surd:

A fraction whose denominator is an irreducible quadratic surd can be converted to an equi-valent fraction with a rational denominator by rationalizing the denominator, i.e. multi-plying both numerator and denominator by the original denominator.


h A k A h k A+ = +( )

627

69 3

63 3

23

23

33

2 33

=)

=)

= = ) =

42

42

22

4 22

2 2= ) = =

2 3 4 3 6 2 2 3 6 2( + = ( +

3 5 15) =

example


clil math - liceoscacchibari.edu.it › wp-content › uploads › 2014 › 11 › clil-math.pdfclil...

Documents