Vocabulary 3rd

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MATHS VOCABULARY AND ACTIVITIES

3

rd

CSE

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3DUnit 1: 2D SHAPES

Polígono Figura plana Cuadrilátero

= polygon = 2-D shape or plane shape = quadrilateral

Polígono regular Radio Diagonal Apotema

= = = =

Ángulo central Ángulo interior

= central angle = interior angle

Ángulo recto Ángulo agudo Ángulo obtuso

= rightangle = acuteangle = obtuseangle

Linea recta Semirrecta Segmento

= straight line = ray = segment

Rectas paralelas Rectas perpendiculares

= parallellines = perpendicular lines

Circunferencia Diámetro Cuerda Arco

= = = =

Centro (de la circunf.)

= central point

Figuras circulares Círculo Semicírculo

= circular shapes = circle = semicircle

Sector circular

Segmento circular

= circular sector = circular segment

Ángulos complementarios Ángulos suplementarios

= complementary angles: their measures add up to 90 degrees = supplementary angles: their measures add up to 180 degrees

regular polygon radius diagonal apothem

circumference diameter chord arc

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PerĂ­metro Diagonal mayor Diagonal menor Base mayor Base menor

=perimeter =major diagonal =minor diagonal = bigger base = shorter base

Lado

= side

VĂŠrtice Eje de simetrĂ­a Punto medio Longitud Paralelo

= = = = =

Teorema de Thales

= Thales Theorem

Cateto Hipotenusa

= cathetus (pl. catheti) = hypotenuse

1m2 =

Square metre

corner or vertex (pl. vertices) axis of symmetry middlepoint length parallel

Pythagorean Theorem: Example:

Circumference:For English-speaking people, a circumference is the complete distance around a circle. Therefore, what is the length of the circumference for us.This can be a bit confusing.

Radius (radii pl.): A straight line from the centre to a point on the circumference.

Diameter: A straight line going from a point on the circumference through the centre to the opposite point on the circumference. A diameter is twice the length of a radius.

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Chord: A straight line going from a point on the circumference to another and which does not pass through the centre.

Arc: A portion of the circumference.

Circular sector: The area enclosed by two radii of a circle, and the enclosed arc.

Circular segment: The region between a chord of a circle and its associated arc.

Types of Quadrilaterals:

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Centers of a triangle:The main centers of a triangle are:

Altura de un triángulo

height oraltitude of a triangle: An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side.

The three altitudes intersect in a single point, called the orthocenterof the triangle.

Mediatriz

perpendicular bisector: A line which cuts another line into two equal parts at 90°.

The three perpendicular bisectors meet in a single point, the circumcenter.

Bisectriz

angle bisector: The bisector of an angle is the line that divides the angle into two equal parts.

The intersection of the angle bisectors is the incenter.

Mediana

median: A median of a triangle is a straight line through a vertex and the midpoint of the opposite side.

The intersection of the medians is the centroid.

Area of Plane Shapes Triangle

Area = ½ × b × h b = base h = height

Square

Area = a2 a = length of side

Rectangle

Parallelogram

Area = w × h

Area = b × h

w = width h = height

b = base h = height

Trapezoid (US) Trapezium (UK)

Circle

Area = ½(a+b) × h h = height

Area =

π × r2

Circumference = 2 × r = radius

π×

r

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SOLVING PROBLEMS

1.- A painter is on a top of a 35 ft ladder that is leaning against a house. The base of the ladder is 21 ft from the base of the house. If the painter were to fall, how far down would his fall be? Give the result in centimetres.

NOTE:1 ft (feet) = 30.48 centimetres

2.- John wants to put his TV for sale on craigslist. The problem is that John forgot the inches of his TV. TVs are advertised by the inches of the diagonal. John knows the width of the TV is 45in and the height is 28in. How many inches should John advertise his TV as?

3.- In a computer catalog, a computer monitor is listed as being 19 inches. This distance is the diagonal distance across the screen. If the screen measures 10 inches in height, what is the actual width of the screen to the nearest inch?

4.- Two joggers run 8 miles north and then 5 miles west. What is the shortest distance they must travel to return to their starting point?

Give the result in kilometres.

NOTE: 1 mile = 1.6 kilometres.

5.- Oscar's dog house is shaped like a tent. The slanted sides are both 5 feet long and the bottom of the house is 6 feet across. What is the height of his dog house, in feet, at its tallest point?

Give the result in centimetres.

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PYTHAGOREAN THEOREM RAP (Watch it on YouTubehttp://www.youtube.com/watch?v=lb115CBDJew)

In a right triangle this is always true Square the shorter legs, add that’s all you do And, what do they equal? We’ll tell you now: The longer side squared, hypotenuse, wow!

Chorus: So “a” times itself, “a” squared “b” times itself, “b” squared And “c” times itself, “c” squared “a” squared plus “b” squared equals “c” squared.

A long time ago, your in Ancient Greeks A man named Pythagoras, he added a piece, A little piece of history we still use today, The Pyghagorean Theorem, and this is what we say What he found out, a pattern he saw, In a right triangle there was a law, That 2 squares form on the shorter sides, Piece them together, this is what he decides: They equal the square of the longer side line, Named hypotenuse it works every time, So for example, we’ll tell what now, 3 squared plus 4 squared equals 5 squared, pow!

How many squares sounds like a three?

Nine is what we get, we’ll solve a simple lead 4 times itself, 16’s what we get, The sum of the squares, 25 is what we get.

In a right triangle this is always true Name the legs squared, add that’s all you do.

And, what do they equal? We’ll tell you now: Now, the longer side squared, hypotenuse, wow!

Chorus What if I know the longer side’s length and the shorter leg? I’ll use my math strengths with the hypotenuse subtract the leg squared, square root the difference if you dare.

Another example, we’ll give you some The longest leg 10, 8 the shorter one, 100 minus 64, 36 we get Root the 36, 6 we’ll bet.

Chorus In a right triangle this is always true Square the shorter legs, add that’s all you do And, what do they equal? We’ll tell you now: The longer side squared, hypotenuse, wow!

Chorus (Repeat twice)

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Unit 2: 3D SHAPES Cuerpogeométrico

= 3D shape

Poliedro

= polyhedron

Poliedro regular

= platonicsolid

Prisma

= prism

Prisma triangular

= triangular prism

Prisma cuadrangular

= squareprism

Pirámide

= pyramid

Apotema de la pirámide

= slantlenght

Esfera

= sphere

Cilindro

= cylinder

Cono

= cone

Generatriz del cono

= slant height or lateral height

Altura del cono

= altitudeor vertical height

Arista básica

= basic edge

Anchura

= width

Profundidad

= depth

Altura

= height

Área lateral

= lateral area/ side area

Área de la base

= base area / area of the base

Metro cúbico

= cubic metre

1 m3

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ACTIVITIES: 1) Translate into English (you can find them in our vocabulary: Edmodo, blog …); Poliedro = Arista = Cara = Prisma pentagonal = Ortoedro = Pirámide cuadrangular = Cono = Metro cuadrado =

Generatriz del cono= Arista básica = Prisma = Prisma hexagonal = Pirámide = Cilindro = Esfera = Metro cúbico =

Now you have to answer different questions. In order to do so, we are going to visit: http://www.mathsisfun.com/geometry/index.html 2) Define Polyhedron:

3) The five Platonic Solids: Name

Number of faces

Faces (name of the polygons)

4) Prisms: a) What is a cross section?

b) What polygon is the cross section of a pentagonal prism? c) If the cross section were a circle, it wouldn’t be a prism. What would it be?

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5) Pyramids: a) Draw a square pyramid and name its different parts: base, apex, slant height, height, edge, basic edge.

b) What is an irregular pyramid?

6) Non-polyhedra: a) Write three properties of the sphere: 1.2.3.-

b) Name three objects that are cone shaped:

c) Name threecylindrical objects.

d) Draw a cone and name the height, the radius and the side length.

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Unit 3: TRANSFORMATIONS Transformations are the changes in the position or size of a shape.

 Types of transformations: Translation (traslación) Reflection (simetría) Rotation (giro)

We get congruent shapes

Enlargement (semejanza)

We get similar shapes

If two shapes are similar:  They have the same shape  All the corresponding angles are equal  All the corresponding lengths are in the same ratio.

A tessellation is an arrangement of a shape or set of shapes on a flat surface so that all the space is filled, with no gaps and no overlaps.

module Vector direction (English people don´t speak about “sentido”in mathematical terms)

UTENSILS /ju:’tensils/:  Ruler (regla)  Protractor (transportador de ángulos)  Compass (compás. Se usa mucho en plural: compasses) Note that we can say: - Set your compasses to (a radius of) 4 cm - Place your protractor over point P - Mark a point at 100º and join it to P with a line Go to: www.youtube.com/watch?v=0Z1aUhGCZs0 and SING!

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Unit 4:RATIONAL NUMBERS INTEGERS: The opposite of a is -a.

The opposite of -

ais a.

Really we can! We knowINTEGERS!

POWERS AND ROOTS: 42    

four squared .

7     seven cubed 15 4     fifteento the power of four/ to the fourth 3

65289     sixty  five to the power of two hundred and eighty  nine

To square a number is to multiply it by itself.

Write as a single power the square root of twenty  five is five 25  5     Twenty  five square root is five 5

243  3     the fifth root of two hundred and forty  three is three

6

32  2

    the sixth root of thirty  two is two

DIVISIBILITY: 

12 is divisible by 4and12 is multiple of 4and 4 is a factor of 12

 

prime numbersare: 2, 3, 5, 7, 11, 13, 17 … compound numbers are: 4, 6, 8, 9, 10…



9 and 16, 34 and 25 are prime numbers between them.



Write 36 in prime factor form means we write

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FRACTIONS: o Proper fraction: numerator is less thandenominator o Improper fraction: numerator is greater thandenominator

Equivalent fractions: fractions that represent the same number.

Amplify (a fraction) Simplify (a fraction): we can simplify a fraction if the numerator and denominator have a common factor.

A fraction is in its simplest form when it cannot be simplified any more. 

Reading of fractions: 3 = three fourths/three quarters 4 5 = five halves 2 7 = seven ninths 9 2 = two over sixty-five 65

DECIMAL NUMBERS: 

Approximating a decimal number: o by rounding o by truncating.



Percentages: Calculate how many percent…

65%… sixty-five percent of …

Displace the decimal point

to the left or to the right

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TYPES OF DECIMAL NUMBERS THAT WE CAN GET FROM A FRACTION:

Pure recurring decimal Mixed recurring decimal

Reading of decimal numbers: o 1.827

= one point eight two seven

o 35.15

= thirty-five point one five

o 3. 1414…

= three point one four repeating

o 3.14343…

= three point one fourthree with four three repeating

o 3.01111… = Three point zero one with one repeating

Order or hierarchy of the operations: Remember!!

P

Parentheses first

E

Exponents (ie Powers and Roots)

MD Multiplication and Division (left-to-right) AS Addition and Subtraction (left-to-right)

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FINALLY IN THIS UNIT‌. ‌.The different sets of numbers that we are studying:

Set of numbers that can be written as a quotient of two integers: Q

Set of whole numbers and their opposites: Z

Set of natural numbers and the number 0 Set of numbers starting with 1 and counting up by ones: N

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Activities: RATIONAL NUMBERS. I)

Fill in the gaps.

1)

Numbers 1, 2, 3, 4, 5, … are ______________ numbers.

2)

All positive and negative whole numbers

are _____________________

3)

The number 3 is the ____________ of the number -3.

4)

The __________ ____________ of 5 and -5 is 5.

5)

Two times twenty-eight is ____________

6)

Two hundred and eighty-five divided ___ three is ninety-five.

7)

2, 3, 5, 7, 11, 13, 17 … are _________ numbers; 1, 3, 5, 7, 9…..are ____________

numbers and 2, 4, 6, 8, 10, 12… are __________numbers 8)

A ____________ is a number that is expressed in the form p/q where p and q are

integers and q is not equal to zero.

9)

In a fraction, the____________is the number of parts the whole is divided into.

10)

A ___________ number is a number that can be written as a fraction.

11)

1 is in its _______________ _________, because it cannot be simplified any more.

4

12)

6.333… is a _________ ____________ decimal number.

13)

0.25555… is a ___________ ___________ decimal number.

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II)

Complete the text using the given words:

other, whole numbers, zero, add, negative, higher, kinds, numbers, beginning, subtrahend, result, multiply, contains, sign, count, integers, positive, opposite, below.

At the _________ people used _________________ to _________ their goods and identify things, and they were able to ________ and _________those numbers (obtaining ________ whole numbers). They could subtract when the minuend was __________ than the_________________. But people created other _________ of _____________ to solve this, as well as to express debts, temperatures __________ zero… These new numbers are called integers. The set of the integers ___________ the natural numbers, the ______ and the _______________of every whole number (expressed by the _________ “-” before it, like – 6). So the ______________of a sum/multiplication/subtraction of two ____________ is another integer, that can be zero (0), a __________ number (2, 6, 152…) or a ______ number (-24, -85, -652…).

III) Reading Numbers Write, in English, how we read: 7  2 5  39 8.75151515....  56.77777....  3  7 932.111111..  1  4 74.2323 

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IV)

Answer the following questions:

a) What is a rational number?

b) When can we simplify a fraction?

c) Write the number four point one three repeating. d) Is three thousand seven hundred and thirty-one an even number? e) Write the number two hundred and two point zero one seven with one seven repeating. f) Define prime number.

g) Is one hundred and eleven a prime number? h) Write three odd numbers bigger than 40. __________________________

V)

Put the words in order and rewrite the sentences.

a) is less than the numerator is called an improper fraction A fraction when the denominator.

b) the unit (one) numerator and denominator are equal, the fraction When the represents.

c) they represent are equivalent when the same number Two fractions

d) obtain equivalent fractions You can numerator and denominator by the same number by multiplying/dividing.

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SOLVING PROBLEMS 1) For my mother’s birthday we gave her a box of chocolates. We have eaten ¾ of the box. If the box contains 40 pieces of chocolate, how many pieces remain?

2) Three eighths of the students in the high school wear glasses. If 129 students wear glasses, how many students are there in total?

3) A farmer wants to build a fence 2,275 m in length. The first day he built 3/7 of the fence. The second day he built 2/5. How many meters of fence need still to be built?

4) Some friends rode 105 km on bicycle. The first day they went 1/3 of the way, and the second day 4/15. They left the rest for the third day. How many kilometres did they travel each day?

5) A family spent 1/15 of its monthly income on the rent for their apartment, 1/60 on the telephone bill and 1/8 on transportation and clothes. How do they distribute their spending if their monthly income or earnings is 3,000 euros?

6) At a camp, 3/8 of the children are European, 1/5 is Asian, and the rest are African. There are 800 children in total: a. How many European children are there? b. If half of the Asian children are girls, how many Asian girls are there? c. How many of the children are African?

7) Cristina has to read a book for school. The first day she reads a quarter of the book, the second day she reads half of what is left. What fraction represents what she read the second day?

8) We have a piece of wire 90 m long. We sold 2/3 of the wire at 3 euros/m. 1/6 of what was left at 4 euros/m, and the meters that were left at 2 euros/m. How much money will you make if you bought it at 2 €/m?

9) A pool is 7/9 full to its capacity. They need 800 litres to completely fill the pool. What is the total water capacity of the pool?

10) A train has already covered three fifths of its itinerary. Even if there is still 84 kilometres until the end, how far has it gone?

11) Three friends divide 90 euros they had made in a betting pool. The first took one fifth, the second one third of what the first friend received, and the third friend took half of what was received by the second. a. What fraction represents what each friend made? b. How much money did each friend make? c. How much money was in the betting pool?

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Unit 5: REAL NUMBERS A real number is any number on the number line.

Real numbers is the union of the sets of rational numbers and irrational numbers.

So,

TYPES OF DECIMAL NUMBERS:

RATIONAL NUMBERS

IRRATIONAL NUMBERS: are decimal numbers neither recurring nor terminating.

INTERVALS: An interval is the set of all real numbers between two given numbers.

The two numbers on the ends are the endpoints. 

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.

Note: Infinite (adjective) / infinity (noun) For instance: Real numbers is an infinite set.

Infinity is one endpoint of the half-open interval (-∞, -1]

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Activities: REAL NUMBERS. I)

Reading Numbers

Write, in English, how we read: 36  417  3

8

98  5

47 

27 3  52  5.6666....  7.898989....  7  8 9  2 54  3

93 

13 2 

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II)

Fill in the gaps.

1) 2 to the power of five is ______________ 2) Nine squared is ____________________ 3) There are other numbers that cannot be written as a fraction, for instance the square root of 2, and because they are not rational they are called ____________ 4) There are some important irrational numbers in mathematics such as π; one of them is the number e and another is  (the golden ratio). Look for information and answer:

- Write π using six decimal figures ________________________ - Write e using six decimal figures________________________________ - Write  using six decimal figures ___________________________ 5) The two methods of approximating a decimal number are: ________________ and

_______________ 6) The set of all real numbers between two given numbers is an ____________ 7) The two numbers on the ends of an interval are the ______________ of the interval.

8)

6 and

0.23242526... are ___________________ numbers

9) When I write 18 = 2  3 2 , it is said that 18 is written in ________ _________form.

10) If I had to answer this question: Write as a ______________ power the following expression: 2 4  2 6  2 2 , I´d write 212 11) The number 2.3x10-5 is in _____________ ________________ or scientific notation.

12) 3.232425…. is an _____________ number.

13) 6, -4,

1 and  are _________ numbers.

2

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DONALD IN MATHMAGICLAND 1. You can see a bird saying a lot of decimal figures of the number ď ? . This bird has been drawn using geometrical figures. Write the name of one of the three 2-D shapes

that have been used.

2. What famous Greek mathematician studied the connection between music and maths?

3. Draw the emblem of the Pythagoreans?

4. What adjective is used by Donald Duck to refer to people interested in maths?

5. In the film you can see different places, things‌ where you can find the golden rectangle or the golden section. Write two of them. a. b.

6. Name different things in nature that have pentagonal shape.

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7. Who said: ”Everything is arranged according to numbers and mathematical shapes”?

8. Write the name of some games related to mathematics.

9. We can see Donald dress up as the main character of a famous tale.

a. What is the name of this character? b. What is the name of the tale? c. Can you tell me who wrote it? d. What connection had the author with maths? e. What historical period he belongs to?

10. Name some of the 3-D shapes that appear in the film.

11. Try to explain the meaning of the following sentence:

“The mind knows no limits when used properly”

12. You can see a lot of doors that are locked. What is the key that can open them?

13. At the end of the film we can read the sentence: “Mathematics is the alphabet which God has written the universe”. Who said this sentence?

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Unit 6: SEQUENCIES What is a Sequence?

A Sequence is an ordered list of numbers.

For instance: a)

Other examples:

b)

c)

1 , 4, 9, 16, 25 . . .

A Sequence usually has a rule, which is a way to find the value of each term.

In the example b) we can say "starts at 1 and jumps 4 every time”. However, this is not the best way to calculate the: 10th term, 100th term … nth term, where n could be any term number we want.

  

In order to do this, we want a formula with "n" in it.

This formula is called the nth term of the sequence. We often use this special style:  

xn is the term n is the term number

Example: to mention the "5th term" you just write: x5

Arithmetic Sequences In an Arithmetic Sequence (or arithmetic progressions) the difference between one term and the next is a constant.

In other words, you just add some value each time ... on to infinity.

Example: 1, 4, 7, 10, 13 …

This sequence has a difference of 3 between each number.

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In General you could write an arithmetic sequence like this: a, a + d, a + 2d, a + 3d, ... where:  

a is the first term, and d is the difference between the terms (called the "common difference")

And the rule is:

an  a1  (n  1)  d

The Sum to n terms of an AP is:

Sn 

(a1  an )  n 2

Geometric Sequences In a Geometric Sequence (or geometric progression) each term is found by multiplying the previous term by a constant.

Example:

2, 4, 8, 16, 32, 64 …. This sequence has a factor of 2 between each number.

In General you could write a geometric sequence like this:  

a, ar, ar2, ar3, ...

a is the first term, and r is the factor between the terms (called the "common ratio")

Note: r should not be 0.

And the rule is:

an  a1  r n1

The Sum to n terms of a GP is:

an  r  a1 a1  (r n  1) Sn   r 1 r 1

The sum of “all the terms” of a GP with 0  r  1 is

S

a1 1 r

The Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34 ...

The next number is found by adding the two numbers before it. The rule is: an  an1  an2 ; a1  1 , a2  1 Rules like that are called recursive formulas.

where:

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EXERCISES: SEQUENCES 1) What is a sequence?

2) Write the 6th and 7th terms of the following sequences: a) 1, 4, 9, 16, 25 … b) 2, 5, 10, 17, 26… c) -1, 2, 7, 14, 23 …

3) Write the rule for the nth term of the sequences given in the previous exercise.

4) When do we say that a sequence is an arithmetic progression?

5) Write 2 examples of arithmetic sequences, writing the difference of each one.

6) Write the formula of the sum to n terms of an arithmetic progression.

7) Calculate the sum to 15 terms of the progression: 5, 0, -5, -10 …

8) What is a geometric progression?

9) Write 2 examples of geometric sequences, writing the ratio of each one.

10) Is 624 a term in the sequence 4, 10, 16, 22, . . . ?

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Unit 7: 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.

Monomials: 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:  9 y 3 , 2 x z, 3a 2 b, 5



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.



A coefficient is the numeric factor of your monomial.

Examples:



Monomial

Coefficient

Degree

5y3z

5

4

3z

3

1

-6

-6

0

A constant term is a monomial that contains only a number. In other words, there is no variable in a constant term. Ex: -5, 4

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Polynomials: A polynomial is a finite sum of monomials where the exponents on the variables are nonnegative integers. The monomials in a polynomial are called 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  4 x 3  2 x 3  x  9  5 x  3

we will write it in this way:  2 x 3  3x 2  4 x  6 . This is a third- degree polynomial.

And the polynomial



is a fifth-degree polynomial.

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

In the Polynomial: P(x) =

cn

is called the leading coefficient. c0 is the constant term n is the degree

This 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.

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Ruffini's rule: Ruffini’s rule allows the rapid division of any polynomial by a binomial of the form x − r.

Example: Let: P(x) = 2x3 +3x2 - 4 and D(x) = x + 1. Divide P(x):D(x) 1. Write down the coefficients and r. 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 r: | 2 3 0 -4 | -1 | -2 ----|---------------------------| 2

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 {result coefficients}{remainder}

So, the quotient is Q(x) = 2x2 + x – 1 and the remainder is R = - 3 The remainder R is equal to P(r), the value of the polynomial at r (the polynomial remainder theorem)

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EXERCISES: ALGEBRAIC EXPRESSIONS 1.- Write an algebraic expression that will symbolize each of the following sentences: a) Six times the number n. b) Six more than a certain number a . c) Six less than a certain number z. d) Six minus a certain number n. e) A number z repeated as a factor three times. f) The sum of three consecutive whole numbers. g) Eight less than twice a certain number a . h) One more than three times the number n. i) The sum of two numbers is twenty. j) The quotient of two numbers is equal to the sum of those numbers.

2.- a) What is the numerical value of 3(2x - 5) when x = 4? b) What is the numerical value of 6g + 10k when g = 5 and k = 2?

3.- Let x = 10, y = 4, z = 2, and evaluate the following expressions: a) x + 2(y + z) =

b) (x + 2)·(y + z) =

c) x − 3(y − z) =

d) (x − 3)·(y − z) =

4.- Remove parentheses and collect like terms. a) (a + 2b + 4c − 3d) − (3a − 8b − 2c + d) = b)

(5xy − 3x + 2y − 1) − (2xy − 7x − 8y + 6) =

c) (4x² − 7x − 3) − (x² − 4x + 1) = d) (x² + x + 1) + (2x² + 2x + 2) − (x² − x − 1) = 5.- Factor each sum. Pick out the common factor. a) 8x + 12y − 16 = b) 8ab3 + 12a²b² = c) 36y15 − 27y10 − 18y5 = d) 20x4 − 12x3 + 36x² − 4x = e) x6yz² + x²y4z3 − x3y3z4 =

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6.- Describe each polynomial (say variable, degree, constant term and leading coefficient) a) 8x3 − 2x² − 3x − 4 b) 3y² + y5 -2y + 1 c) -8 - 4w 7.- Multiply and simplify the results: a) −(x + 1)(x − 2) = b) 4(x − 1)(x + 3) = c) x(x − 2)(3x + 4) = d) (x + 1)(x + 2)(x + 3)= 8.- Multiply. Cancel first

9.- Remember: To add fractions with different denominators, we must use the Lowest

Common Multiple of the denominators. Calculate: a) b)

3 2 − x−3 x 6 3 + x−1 x+1

c) 2x − 3 x − 4 − x+2 x−2 10.- Fill in the gaps with the right word: a) If we substitute letters for _____________ in an algebraic expression and carry out the operation indicated, we get its _______________ b) The ______________of a monomial is the sum of the exponents on the variables contained in it. c) A ___________________ is an algebraic expression where the only operations indicated with the letters are products and powers (with natural numbers in the exponents)

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Unit 8: EQUATIONS Igualdad

= equality

Identidad

= identity

Ecuación de primer grado

=a linear equation

Ecuación de segundo grado

= a quadratic equation /a second-degree equation

Ecuación de grado 3

= a third-degree equation

Miembro de una ecuación

= equation side

Primer miembro (de una ecuación)

= left-hand side (of an equation)

Segundo miembro

= right-hand side

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

Equation in Words Seven more than a number is negative ten.

Equation in Symbols 7+ n = -10

A number plus two is eight.

x+2=8

The difference between a number and seven is negative three.

x – 7 = -3

One less than twice a number is seventeen.

2x – 1 = 17

A number increased by eight is 87.

n + 8 = 87

Half of a number is twenty.

Twice a number, decreased by 3, is 42.

= 20 2x -3 = 42

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SOLVING PROBLEMS WITH THE HELP OF EQUATIONS 1) A number and the same number minus one add up to 77. What are the two numbers?

Solution: 39 and 38

2) If you add thirteen to two times a number, you obtain a result of 99. What is the number?

Solution: 43

3) In a pride of 13 lions, there are 3 more females than there are males. How many lions and how many lionesses are there?

Solution: 5 and 8 4) There are 31 people in a café. How many men and how many women are in the café if there are 5 more men than women?

Solution: 13 and 18 5) In a farm, the horns and legs on the cows add up to 30. How many cows are there in the farm? Solution: 5 6) Mark bought two pens and a marker for a total of 5 euros. What was the price of each of the items if a marker costs fifty cents more than a pen? Solution: 1.50 € and 2 € 7) The base of a rectangle is three centimetres longer than the height of the rectangle. Its perimeter is 38 centimetres. What are the dimensions of the rectangle? Solution: 11cm and 8 cm 8) A box of figs weighs one kilo more than a box of strawberries. Together, three boxes of strawberries and two boxes of figs weigh 12 kg. How much does each box weigh?

Solution: 2 kg and 3 kg

9) A bowl of ice cream costs eighty cents more than a pasty. At snack-time, Mary and Lisa bought one bowl of ice cream and two pasties for a total of 4.40 €. How much does a pasty cost? How much does a bowl of ice cream cost? Solution: 1.2 € and 2 € 10) Calculate the dimensions of a rectangular plot of land, given that the plot’s length is 20 metres more than its width and the fence that surrounds the plot is 240 metres long.

Solution: 50 m and 70 m

11) Gus is ten years older than his brother, and six years from now he will be twice his brother’s age then. How old is Gus now? Solution: 14 years old 12) Doreen is five years younger than her brother is and three years ago the sum of their ages was 23 years. How old is each now? Solution: 12 and 17 years old 13) Tom has 150 feet of fence to enclose a rectangular garden. If the length is to be 5 feet less than 3 times the width, find the area of the garden. Solution: 1100 square feet

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Unit 9: SYSTEMS OF LINEAR EQUATIONS 

Two equations using the same variables for which you need a common solution are called a system of equations or a pair of simultaneous equations.

Substitution method for solving systems of equations: 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.

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Elimination method (or addition method) for solving systems of equations: 1.- Multiply every equation by the properly number in order to get one of the unknowns with the same coefficient in both equations.

Don’t forget! You have to multiply both sides of the equation by the chosen number.

2.- Add the two equations.

3.- Solve the equation that you get.

4.- Substitute the found unknown by the value that you have got in one of the equations and solve for the other unknown.

Graphing method: We have to graph both lines and find where they meet.

Example: Solve the system {

So the solution is: x = 2 y=2  Number of solutions of a linear system of equations:

The lines are parallel

The lines intersect

The lines coincide

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SOLVING PROBLEMS WITH SYSTEMS OF EQUATIONS 1.- Twice one integer plus another integer is 21. If the first integer plus three times the second is 33, find the integers.

3.- In a corral (farmyard) there are 40 animals split between chickens and rabbits. The total amount of legs is 106, how many rabbits and how many chickens are there?

4.- Three times the age I was 2 years ago is double the age I will be in 6 years. What is my current age?

5.- I have 15 coins, some are 5 cents and others are 10 cents. If the total amount is 1.40 â‚Ź, how many 5 cents coins and how many 10 cents coins do I have?

6.- Two numbers add up to 32 and their quotient is 3. What are these 2 numbers?

7.- The age of a boy is 4 times younger than his father´s age and 6 years ago he was 7 times younger. How old is the father and how old is the boy?

8.- An installer of underground irrigation systems wants to cut a 20-foot length of plastic tubing into two pieces. The longest piece is to be 2 feet longer than twice the shortest piece. Find the length of each piece.

9.- There are two boxes in the square. One contains green balls, and the other, orange balls. There are 13 balls in total. Then, 5 green balls and 2 orange balls are added, so that the number of green balls is triple the number of orange balls. Find out how many balls of each colour there were at the start.

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Unit 10: FUNCTIONS AND GRAPHS A function is a relationship between two variables that assigns to each value of the first one and only one value of the second.

Independent variable

Dependent variable

It is not a function because a vertical line crosses the graph more than once.

Note: The plural of axis is axes.

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

The range of f is the set of “output� that the function takes when x takes values in the domain.

Table of values: A table of values is a list of numbers that are used to substitute one variable to find the value of the other variable.

For example:

Input x -1 0 1 2

Output Y=2x -2 0 2 4

We calculate y by substituting the xvalue into the formula of the function

We get ordered pairs: (-1,-2) (0 , 0) (1 , 2) (2 , 4)

x and y are the coordinates of this point

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Monotony of a function: Increasing function

Decreasing function



A function is "increasing" if the y-value increases as the x-value increases.



A function is "decreasing" if the y-value decreases as the x-value increases.

Maximum and minimum points: The graph of a function has a local maximum at one point where the graph changes from increasing to decreasing.

The graph has a local minimum at one point where the graph changes from decreasing to increasing.

Periodicity: A periodic function is a function that repeats its values in regular intervals or periods.

P is named period.

Continuity: A function is said as being continuous if you can sketch the graph of the function without ever lifting your pencil.

Continuous function

Discontinuous function

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STUDYING FUNCTIONS

For every one of the following functions, write: Domain Range x- and y- intercepts increasing intervals decreasing intervals maximum points minimum points Decide if they are continuous or discontinuous functions.

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Unit 11: LINEAR FUNCTIONS

A linear function is a function that can be graphically represented in the Cartesian coordinate plane by a straight line.

In the formula of a linear function the input is raised only to the first power.

(m and b are constants) The slope (also called gradient) of a straight line shows how steep a straight line is.

The gradient (m) is defined as the ratio of the vertical change between two points, to the horizontal change between the same two points.

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

increasing

When the gradient is 0, it is a Constant Function. ď&#x201A;ˇ ď&#x201A;ˇ

Its formula is f(x) = c.

Its graph is a horizontal line.

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EXERCISES: FUNCTIONS 1) Identify the slope and y-intercept of the graph of the given linear functions: a)f(x) = 2x + 1

b) f(x) = -3x + 7

c) f(x) = 9 – 5x

2) Set up a coordinate system on your sheet of paper. Label and scale each axis and draw the graph of the given functions. Remember to draw all lines with a ruler. a) f(x) = 3 – x

b) f(x) =

x–7

c) f(x) =

3) Find the formula for the line that: a) through (2 , 3) with slope 4

b) through (5, -2) and (1 , -2)

4) For the function f(x)= ax + b, f(0) = 3 and f(1) = 4. Write the equation and represent it graphically.

5) Three pounds of squid can be purchased at the market for $18. Determine the equation and represent the function that defines the cost of squid based on weight.

6) It has been observed that a particular plant's growth is directly proportional to time. It measured 2 cm when it arrived at the nursery and 2.5 cm exactly one week later. If the plant continues to grow at this rate, determine the function that represents the plant's growth and graph it.

7) A car rental charge is $100 per day plus $0.30 per mile travelled. a) Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. b) If a total of 300 miles was travelled in one day, how much is the rental company going to receive as a payment?

8) When digging into the earth, the temperature rises according to the following linear equation: t = 15 + 0.01 h

t is the increase in temperature in degrees and h is the depth in meters.

Calculate: a) What the temperature will be at 100 m depth? b) Based on this equation, at what depth would there be a temperature of 100 ºC? 9) The pollution level in the centre of a city at 6 am is 30 parts per million and it grows in linear fashion by 25 parts per million every hour. If y is pollution and t is time elapsed after 6 am, determine: a) The equation that relates y with t. b) The pollution level at 4 o'clock in the afternoon.

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Unit 12: STATISTICS

Statistics is the branch of mathematics that deals with the collection, organization, analysis and interpretation of data.

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.

BASIC CONCEPTS: 

Population is the entire group we

are interested in, which we wish to describe or draw conclusions about. 

A sample is a group of units

selected from the population.



Unit is one of the elements in a population or sample.



Size is the number of elements of the population or sample.

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GRAPHS: BAR CHART

FREQUENCY POLYGON

HISTOGRAM

PIE CHART

CARTOGRAM

PICTOGRAM

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Table of Frequencies CENTRALIZATION MEASURES



xi

fi

hi

Fi

Hi

%

̅ sample mean or average1

o

o Me median 3

PERCENTAGE

RELATIVE CUMULATIVE FREQUENCY

ABSOLUTE CUMULATIVE FREQUENCY

RELATIVE FREQUENCY

FREQUENCY ABSOLUTE

or

DATA MID-INTERVAL VALUE

o Mo mode 2

MEASURES OF DISPERSION



o range 4 o o

variance 5 standard deviation 6

o CV coefficient of variation

1

The mean is the sum of the values divided by the number of values.

2

The mode is the value that occurs most often.

7

3

The median is the middle value or the mean of the two middle values when the values are put in order of size.

4

The range is the highest value minus the lowest value.

5

The variance of a sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the mean.

6

The standard deviation is the positive square root of the variance.

The coefficient of variation represents the ratio of the standard deviation to the mean. We use it for comparing the degree of variation from one data series to another.

7

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Unit 13: PROBABILITY 

A Random experiment is 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: any collection of outcomes of an experiment, any subset of the sample space.



Any event which consists of a single outcome in the sample space is called an elementary 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 the sample space.



Impossible event: It is the event containing no outcomes. It is denoted by Ø.

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

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 possible outcomes

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)

Then: p(Head) = 1/2

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Independent and dependent events: 

Independent Events: These are events that are not dependent on what occurred previously.

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: The chances don't change.

Without Replacement: The chances will change.

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

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.