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DEMO

N O I T A C U D E Y R A D EC O N

s c i t a m e h t a M c i m e d a c A r o f s e i d Stu ez Gonzál a r i e v Oli as , M .ª J . z e n ra Cañ é e l m i o J C . a r ,R J. Cole Albero u l e t z I. Ga

Building

Blocks

THIS IS

YOUR BOOK F EACH UNIT

Reading and listening

THE OPENING PAGES O

We read and listen a brief historical introduction of the contents you are going to learn in the unit.

ns on the money or possessio

‘risk with the text that means: to bet’. Write a sentence 1 Find a word in g which is not certain,

result of somethin its meaning is clear. the word in which thy’ the same as ‘notewor g words does not mean 2 Which of the followin text)? d) Significant (underlined in the c) Outstanding b) Common a) Remarkable text: g questions about the followin the Answer 3 Book on Games of Chance? a) Who wrote The of genetics? ity theory to the study b) Who applied probabil theory? birth of probability c) What led to the ity theory? ons of analytical probabil d) Who led the foundati

chance and probability g

Reading and listenin

chanc betting. The related to games and by Italian theory was closely At first, probability of chance were done , was studies about games Gerolamo Cardano first mathematical th century. One of them, systematic algebraists in the 16 the first somewhat g and for writing famous for his gamblin Games of Chance.

Early games of

The Book on in Cardano’s work on this topic: problems that appears version of one of the Here is a simplified total by book: die, or getting eight a four by tossing one ‘Which is easier, getting

Solve ’s 1 a) Solve Cardano 4

tossing two dice?’

P[4] = …

Probability theory

s that series of chance problemSeveral r de Meré posed a In 1654, the Chevalie the French mathematician Blaise Pascal. Pierre g to mathematician friend were very interestin the problems with his exchange days later, Pascal shared s a different way. Their solved the problem Fermat, and they each s led to the birth of probability theory. of ideas and new problem Meré gave Pascal was: s that Chevalier de The person One of the problem in a game of chance. to 1, the B, bet 3 000 coins 2 ‘Two players, A and takes the money. When A is winning divide the who wins three games is not resumed later. How should they and game is interrupted coins?’ 3 000

Probability as

below: page using the figure problem on the previous die, there is 1 8 • When we toss one a four: possibility in 6 of getting dice, there are • When we toss two of getting an 5 possibilities in 36 eight: P[8] = …

(2, 1) 4th game 1 1 — — 2 2 B wins A wins (2, 2) 1) (3, 5th game 1 1 — — 2 2 B wins A wins (2, 3) (3, 2)

a science

deeper aticians gained a Fermat, other mathem most noteworthy were the The After Pascal and imon new field of study. understanding of this li (1654-1705) and the French Pierre-S ring), is ndi (Art of Conjectu Swiss Jacques Bernoul Bernoulli’s Ars Conjecta ities. Laplace Laplace (1749-1827). published about calculating probabil work and gave probability theory ity the first important of analytical probabil ons’) a huge laid the foundations sense reduced to calculati n ‘commo called theory (which he of Probabilities. work Analytic Theory push forward in his t Gregor Mendel applied th the Austrian naturalis In the mid-19 century,of genetics. probability to the study

1 — 2 A wins

1 1 of — — 2 2

1 — 4 A wins

1 — 4 B wins

, ‘getting 5’ with one

b) What is more probable 7’ with two dice?

die or ‘getting a total

the look at the figure on

problem, Chevalier de Meré’s cases, and A wins in 2 a) To solve the in half of half of the

B wins left. We can see that So: the rest of the cases. 1 P[B wins] = 4 1 1 = 3 P[A wins] = 2 + 4 4 proportionally to is to divide the money The most logical solution Do it. ities. ted probabil these if the game was interrup divide the 3 000 coins b) How would you 2 – 0? when A was winning

GE BANK BANK LANGUA LANGUAGE NK GE BA NK GE BANK GE BA LANGUA LANGUA LANGUA BANK BANK 297 GE GE NK GE BANKNGUAGE BA LANGUA LANGUA UA NG LA BANK LA GE BANK NK GE UAGE BA LANGUA LANGUA

Solve. You can do these motivating activities to activate your previous knowledge.

LANG

296

The audios of each unit’s content are available at www.anayaeducacion.es

CONTENT DEVELOPMENT AND ACTIVITIES Unit

Each unit is divided into epigraphs and subepigraphs. The most important contents are in bold. The icons included with some activities indicate the keys to the project.

1 Terminology The expressions on both sides of the = sign are called sides. In the equation on the right, x + (x + 1) + (x + 2) is the left side and 33 is the right side.

Sequences numbers given in integers a set order. Foradd example: The heightsare of sets threeoftrees are consecutive which up to 33. Find the height of the a) 1, 5,shortest 9, 13,tree. 17… b) 1, 4, 9, 16, 25, 36… We translate statement into algebraic to 81, produce an equation: c) 2,can 4, 8, 16, 32,the 64… d) 1, language –3, 9, –27, –243… + (x (x + 2) = 33 e) 1, 1,x 2, 3, +5,1)8,+13… f ) 1, 2, 4, 7, 11, 16…

This means that: want x + (x + 1) + (x + 2) to equal value must x g) 2, 6, 12, 20, 30,‘We 42… h) 170, 120, 70,33’. 20, What –30, –80… be to achieve this? i) 1, 3, 6, 8, 16, 18, 36… j) 2, 3, 5, 7, 11, 13, 17, 19… Solving an equation is the same as answering that question. Stating that the Each of the is saying created‘when by following particular solution is x sequences = 10 is theabove same as x equals a10’, it is truerule. that:Some of them are obvious: x + (x + 1) + (x + 2) equals 33 — The next term is obtained by always adding (or multiplying by) a fixed amount. An equation, rather than an equality, is an equality statement that contains

position

1st 2nd 3rd …

term

a1 1

example

Sometimes, is possible to find expression that allows us to findmethods. any term In in In previous it years, you learnt to an solve equations using algorithmic aother sequence just bylearnt knowing its position. words, you to follow a series of steps that lead to a solution. In the following sections of this unit, we a) will on 17… using from these the methods to page. solve For example, let’s look at sequence 1, insist 5, 9, 13, previous first-can andfind second-degree equations. We the expression a = 4n – 3 because if we give n the values 1, 2, 3,

nth

a2 a3 … an 5 9 … 4n – 3

anayaeducacion.es Reinforce your understanding of sequences.

, a4find ... the solution to an equation using trial 4…, we obtain the terms a1, a2, a3to However, it is perfectly acceptable aand= error. 4n – 3 is the general term of the sequence. n

Sometimes, we rely on* trial and error (with or without a calculator) because we think will be easy ‘see’ the solution. ocasions, trial andthe error will be The itexpression thattorepresents any termIninother a sequence, s, is called general a term most of acceptable andand easy way of solving a sequence it is written as sn. a particular equation. If the solution is an integer, it is easier to find. If it is not, we must repeatedly apply trial and The general term of some sequences can be expressed using the following error to find a decimal approximation. In these cases, we must use a calculator. formula: sn = f(n). If we know the value of n, we can obtain the corresponding Here are some examples: term.

F ocus on English rely on: to trust something to do what you need or expect it to do. anayaeducacion.es Reinforce your understanding of the general term of a sequence.

The sequences that we will be using this year all follow a specific rule. Some of these sequences will be given using their general term or we will be able to find Problem solved it easily. For other sequences, you will have to operate with the previous terms to Solve through trial and error: find thethese next equations ones.

letter.term We call thissquare letter of theitsunknown. —a Each is the position in the sequence. The solution equation Other patterns to arean less obvious:is a value of the unknown that makes the equality true. — Each term is obtained by adding the previous two terms together. Solving an equation means finding the solution or solutions. Sometimes, this —can Alternatively, we can add or multiply mean realising there is no solution. by the same number.

Which sequences on the right correspond to these images?

4 6

ã General term of a sequence Solving equations through trial and error

SEQUENCES EQUATIONS. SOLVING AN EQUATION

a) x x = 3 125 b) x 6 = 1 200 When we obtain the terms of a sequence based on the previous ones, we say a) find the solution quickly if we apply trial and error to integers thatWe it iscan a recurrent sequence. x = 5 so 55 = 3 125.

— The terms are the prime numbers written in order.

Your teacher knows best

TheTypes numbersof in aequations sequence are called the terms of a sequence. We can refer to ã

Solving equations through guesswork For in integer sequence e) 1, 1,to2,x,3,we 5, notice 8, 13…, each term is the sum of the b) example, If we assign numbers that: is very educational because it improves previous two terms. It is defined as: our mental arithmetic skills, even if 3 6 = 729 Therefore, less than 4. e1 =x1,is greater e2 = 1, than en = e3n and we use a calculator. 4 – 1 + en – 2 4 6 = 4 096 In other words, x = 3… However, your teacher will tell you whether or not it is best to use this If we assign the values 3.1, 3.2, 3.3… to x, we can see that: method. They practise always know what is Think and best for your education! 3.2 6 = 1073… < 1200 x = 3.2… 4 Therefore, 6 = 1291… >10 5 Look at sequences b), c), d) and h) on the Find the recurrence relation and add a new term to 3.3previous 1200

the first term, the second term, the third term… of a sequence, s. This can be There are many simplified to: s1, different s2, s3… types of equations. For example: • Polynomial equations. The unknown only expressions. This way, for example, in the first sequence, a1,appears a2, a3...,intopolynomial state that the difference These equations: between each term and the one before it is 4, we can write: = a4+–3)a3 = …x=3 4– 9x = (x + 1)2 + 3 a2 – a1 =x 2a3––2 a=2 2(x 3(x – 5) + x – 3 = 15 2 aresequence first-, secondand polynomial equations, respectively. A is a set ofthird-degree numbers written in order so that each number has a position: first, second, third… • Radical equations. x + 17 + 2 = x – 1 The elements of a sequence are called terms. They are usually expressed as a 3 = 1 The subscript number tells x + 2 in by a number written – subscript. • letter With followed x in the denominator. x +sequence: 3 x –1 8 us the position of the term in the • With x in the exponent. 2sx1,= s64; = 3 125 , x s=4, 81; s5, xs6x… 2, s33 There are many other types of equations. However, this unit will only focus on first- and second-degree polynomial equations.

Think and and practise practise Think 1 Find the rule for each of these sequences: 1 a) 1/2, Is 2/3, 5 the3/4… solution to any of the following equations?

Justify your answer. b) 3, 4, 5, 6… a) 8x + 3 = 11x – 12 b) x 4 – x 3 = 500 c) 1, 8, 27… d) 1x = 5 c) 3x – 7 = x 2 – 10 2 Find the rule for each of the sequences above and add 12 =terms 4x – 7to each one. f ) 2x – 1 = 16 e) x 2 –more three

3 Make five sequences by following rules that are similar

3 + x 2 + 2x + 1 = 161 h) 10x + 25 = x 3 g) to xthose above. Make your own rules for some of the sequences. i) x 2 – 20 = 2x – 5 j) 3x + 1 = 16 2

c + c3) – 84 = 0 l) 3(x k) (2x – 3) = 144 4 State which is the relationship 2 = 3 = … between

c1 c2 2 The exercise terms contains several polynomial each previous two consecutive in sequences c) and d) equations. Write them down and indicate their degree. above.

Focus on English. Do you think Mathematics and English haveanything in common? Discover how language and mathematics are linked so you can learn both: Mathematics and English.

page. Check that: bn = n 2; cn = 2n; dn = (–3)n – 1; each of the following sequences: We can say that the solution to the equation, rounded to the nearest tenth, is hn = 220 – 50n. a) 1, – 4, 5, –9, 14, –23… (Difference) x = 3.2. b) 1, 2, 3, 6, 11, 20… (Connect each term to the previous three) an = n 3 bn = n 2 – 3n + 7 cn = n – 3 n+4 c) 1; 2; 1.5; 1.75… (Semi-sum)

Examples and solved problems. To put into practice the most important methods.

6 Write the first five terms of:

Think and practise 7 Use the information below to make a recurrent sequence: 3 Use trialj and solution of the = 2 error jto=find 3 thej integer =j +j

following1 equations:2

n–1

n–2

8 a) 2x 2Invent = 50 two more recurrent b) 2x 3sequences + x 2 = 20using data

differentxto the data above. c) 4 · 10 = 40 000 d) (x – 12)4 = 81 – 6) =four 9 Write e) (3 + the x)(x first 121 terms of f ) 3the x –sequences 23 = 2 with the following general terms: g) x 3 + x 2 = 150 h) 3x = 2 187n – 1 1 5(n – 1) b) a) i) axnx == 346+656 j) bn7x= +3c42=m 9 2 –=nx – 8 k) c5 x =+ 1(n= –151)(n 625– 2) l) d x =– n12 c) d) n

d) 1, 2, 2, 1, 1/2, 1/2, 1… (Quotient)

4 Solve the following equations. Round your answers to 11 the Make a sequence thatthe follows relation: nearest tenth. Use trial the andrecurrence error method with

n = an – 1 + n. You can use any number as the first a acalculator. term. b) x 3 + 1 = 100 a) x 2 = 1 000 12 c)a)x Check that the general d) term –1, 5 = 1 500 x 6 –for40the = 1sequence 460 1, –1, 1, –1, 1, … is sn = (–1)n. e) (x – 3)4 = 35 027 f ) x 4 + x 2 = 40 b) Find the general term of the following sequences: 3 – x 2 = 200 g) x 3 + x 2 = 200 h) x–1, 1, –1… a → 1, –1, 1, n

x+1 i) x 2 – x = b5 → 1, –2, j) 3, x– 4, 5,=–250 6… n

69 111

68 110

Think and practice. These are exercises to apply the theory you have learnt.

KEYS

PROJECT

SDG SDG Commitment Discover the Sustainable Development Goals and be an active part of our commitment to make a more equal and liveable world.

Developing thinking Work on strategies for thinking: reflect on the content you are learning, generate ideas, organise them, debate them, explain them…

Cooperative learning Get involved in your learning and participate in the group’s learning; you will find that cooperating improves performance and harmony in the class.

Emotional education Get to know yourself; identify the situations that bring up complicated emotions and manage them with constructive, self-affirming experiences.

EXERCISES AND PROBLEMS ed

to choose Remember for your portfolio

PROBLEMS solv

At A, they are stopped. The ride begins to move and goes at a constant speed up to the first peak, C. At that point, it starts to fall and the speed increases until it reaches the lowest point, E. From there, the ride loses speed up to the second peak, G, and from there it drops and quickly gains speed again. When it reaches H, the brakes activate, slowing it down to its end point, I.

D E

breaks

speed

Interpreting graphs

Ana sets out at 10:00 to climb a mountain and return to the start point on the same route. This graph shows the distance she walked on her hike:

distance travelled

breaks

a) There are the same number at the point where the two lines intersect; in other words, halfway through 2005 there were around the same number of pre-paid and post-paid contracts.

10 00

year

a) What year were there the same number of pre-paid and post-paid contracts?

Your turn. Exercises and problems to practice the strategies you have just learnt.

Your turn The graph on the right shows the evolution in the popularity of different media over time.

b) What is the long-term behaviour of the difference between pre-paid and postpaid contracts?

Newspapers

100

b) At 6 seconds is the light on or off? And at 7 seconds?

Magazines

c) Will the light be on or off at 15 seconds? And at one minute?

Internet

a) What years is the Internet the same as the others?

60 40

b) How has the Internet changed compared to the others?

c) Analyse both graphs together.

The following graph shows the distance a car travels from when the driver presses the brake until it stops, based on its speed:

1 2 3 4 5 6 7 8 9 10 11 12 13 time (s)

a) How often does the pattern repeat? In other words, what is the period of this function?

Media introduction in Spain (1998-2016) (% of the population)

20 0

5 000

William leaves his house running to not be late to class. Halfway there, he stops to rest for a bit and continues the rest of the way walking. State which graph shows the distance he travels.

distance (m)

4 000

Mercury takes 88 distance 100 (million km) days to complete one orbit around the Sun. Its distance from the Sun 50 ranges between 70 and 46 million kilometres. time (days) This graph shows its 25 50 75 100 distance from the Sun: a) Copy and complete the graph for 300 days. b) Estimate its distance from the Sun in two Earth years. c) When the graph starts, Mercury is 46 million kilometres from the Sun. How much time passes until it is 60 million kilometres from the Sun?

light

c) In the year 2000, there were more pre-paid contracts. The difference got bigger over the next three years, but in 2004 it started to stabilise as the post-paid growth rate remained constant. This tendency continued until 2012, when pre-paid contracts start to drop, decreasing constantly until 2017. Meanwhile, post-paid contracts continue to grow steadily.

3 000

This graph shows a lighthouse switching on and off several times in a unique pattern (each lighthouse has its own): darkness

b) The difference between post-paid and the pre-paid grows over time: pre-paid contracts start to disappear and post-paid contracts keep increasing.

n.° of contracts (in millions) 40

2 000

a) What is the approximate temperature at a depth of 1 500 km? And at 3 000 km? b) Estimate the temperature at the centre of the Earth (6 371 km). c) If the temperature grows constantly by 20 °C/km in the first section, at what depth will it reach 1 000 °C? And 2 000 °C? d) Iron melts at 1 538 °C. At what depth can we be completely sure that iron is in its liquid state?

time (h)

e) Explain why the graph is never decreasing. 2

2 Interpreting two graphs on the same grid

These graphs show the number of pre-paid (blue) and post-paid (red) contracts over the years in a country:

d) At what time interval does she walk down the steepest part?

1 000

depth (km)

c) How long did she walk before stopping to rest? How long did she rest?

1000

b) When did she reach the peak?

Draw an approximate graph of the function distance travelled - speed for the roller coaster on the right.

2000

a) How long does the hike last? What time did she finish hiking?

F engine

Your turn

4000

3000

distance

Draw the graph of the function that relates the speed to the distance travelled along the track. You have to bear in mind that between A and C, the ride is propelled by an engine; and between H and I, the breaks are activated to stop the ride.

The temperature of the Earth increases as a function of depth. At first, it increases quickly and then, it gradually stabilises. The graph shows an estimation of this variation: 5 000

distance travelled (km) 20 16 12 8 4

engine

a) Approximately, how many metres does a car going 55 km/h travel after braking? b) What speed would a vehicle be going if it needed 50 m to brake?

time

distance

C B

Exercises and problems. To put into practice all the knowledge acquired throughout the unit.

Unit 8

distance

unit

problems 4

distance

Carla and Luke rode a roller coaster that looked like this:

Exercises and problems solved. Methods, suggestions, tricks and thinking strategies that will come in handy to solve similar problems.

this resources from

Exercises and

Practise

temperature (°C)

exercises and 1 Building graphs

time

The exercises are divided into topics. Each one is also marked with its degree of difficulty, from one to three.

40 20 10

speed of the car (km/h)

163

162

MATHS WORKSHOP Unit 8

Maths worksho THINK AND DECIDE

Here, you will find readings, activities, advice, information...

PRACTICE MAKES PERFECT!

As a function of time

III

glossary

b) If you now have one 7-litre jug and one 5-litre jar, what would you do to measure 4 litres of water? c) And how would you measure 3 litres of water if you had one 9-litre jar and one 5-litre jar?

III

SELF-ASSESSMENT

Trust in your skills and knowledge, develop creativity, adapt to changing situations and have a proactive and responsible attitude.

anayaeducacion.es Answer key.

1 This graph shows the altitude above sea level that

Ana and Michael reach climbing a mountain: altitude (m)

cube root

The absolute value of the difference between a real value and a rounded value.

OBSERVE AND GRAPH

algebraic fraction

A given quotient of two polynomials.

Non-adjacent angle that is outside the parallel lines.

alternate interior angle

Non-adjacent angle that is inside the parallel lines.

analytic expression of a function

An equation that algebraically relates the two variables.

cumulative frequency decimal number

Karst spring

For each case, draw the graph relating the height of the water in the container decreasing function with the time that passes: a) b) dependent variable

angle bisector

The locus of the points equidistant from the sides of an angle.

apothem

A perpendicular segment from the centre of a triangle to its side.

area

The size of a surface.

arithmetic arithmetic progression axial symmetry

dispersion parameters A symmetry in which the points of one figure coincide note: Beforewith you the startpoints doing of theanother graph for section b), find out: What is a figure when reflected across a line. domain of a function Karst spring?

VI 1 100 1 000 900 800 700 600 An amount that can be multiplied by itself three times500 to result in the number. 400 of all the previous values, The sum of the frequency of a value with the frequencies 300 when the values are ordered from smallest to largest. 200 100 The result of a non-exact quotient. It has an integer portion and a decimal portion separated by a decimal point. 1 2 3 4 5 6 7 8

A function where the dependent variable y decreases as the independent variable x increases.

a) What are the variables involved? What scale is

The variable represented on the vertical axis. It is theused y variable. for each variable? What is the domain of this

function? A type of motion that maintains the direction of rotation, also called a shift.

The branch of mathematics that studies numbers and the operations that we can do with them.

discontinuous function

A function where the independent variable moves in jumps, creating gaps the altitude did they b) How long was the hike? Atinwhat graph. start walking? What was the maximum altitude

A sequence in which the next term is determined by adding the same number (positive or negative), called the difference, to the previous term.

discrete quantitative variable

ellipse

A graphic representation formed of thin 168 bars, where the height of each bar is proportional to the frequency of that value. They are used to represent tables of discrete quantitative or qualitative variables.

equalisation method

c) When did they climb the fastest? When did they

the fastest? Parameters that tell us how far away from the centredescend the values in a distribution are. The set of x values that have y values.

biquadratic equation

Fourth-degree equations that have no terms with odd degrees.

box-and-whisker plot

A graphic representation that visually describes the position parameters (median and quartiles).

case

It takes 10 minutes to empty. When it is empty, a mechanism fills it in 2 minutes. a) Graph the function time-amount of water. b) Is the function periodic? Explain your answer. c) During the first half hour, when is it full? And when is it empty?

describing the relationship between the height, h, of a ball thrown up in the air, and the time, t. Which one is it? A h = 8t – t 2 B h = 40t – 5t 2 C h = – 4t 2 + 80t height (m) 80 60 40 20

they reached? When did they stop for lunch? A variable that can only take certain numerical values.

Water in Fontaine de Vaucluse.

An imaginary line that divides a figure, a shape, or any other thing into two equal and symmetric parts.

bar chart

2 A tank contains 5 L of water to spray on a terrace.

d) Describe their hike.

An equality of algebraic expressions that is only true for certain values of the letters.

equation

An equality statement that includes a letter called an unknown.

equivalent fractions

Two fractions are equivalent when they have the same numerical value.

Every possible result of a random experiment.

equivalent systems

Systems of equations that have the same solution.

central angle in a circle

An angle with its vertex at the centre of the circumference.

exact root

The square root of a perfect square.

chance

A combination of circumstances that supposedly cause an unpredictable event.

factorising polynomials

circumference

A closed two-dimensional curve in which all points are equidistant from the centre.

A method for decomposing a polynomial into the product of other polynomials with the lowest possible degree.

class midpoint

The central value of each interval.

final value

The product of an original value and a variation index.

compatible determinate system

A system that has one unique solution. Its graphic representation is two lines that intersect at one point.

fraction

A given division. It is the result of dividing an integer into equal parts.

frequency polygon

compatible indeterminate system

A system that has infinite solutions. Its graphic representation is two coinciding lines that share all their points.

A graphic representation formed by connecting the ends of the bars in a bar graph or the midpoints of the rectangles in a histogram. It is used to represent quantitative variables.

compound experiment

An experiment that includes more than one random experiment.

frieze

Longitudinal decoration that contains a pattern that repeats in translations.

compound interest

The profit generated over a certain time by adding interest to the original capital.

function

A relationship between two variables that we usually call x and y.

conic section

The curves resulting from different intersections between a cone and a plane.

general term

An expression that represents any term in a sequence.

continuous function

A function without any discontinuities. Its graph can be drawn without lifting the pencil from the paper.

geographic coordinates

A reference system for specifying the location of any point on the Earth. There are two: latitude and longitude.

continuous quantitative statistical variable

A variable that can take any numerical value in an interval.

geometric progression

A sequence in which the next term is determined by multiplying the previous term by a fixed number, called the ratio.

coordinates

A system of values, references... used to determine the position of a point on a plane.

geometric transformation

corresponding angles

Angles that share one side that intersects two parallel lines formed by their other sides.

A transformation in which every point of one figure corresponds to another point of another figure.

geometry

The study of the properties and measurements of figures in the plane or in space.

time (s)

a) What height does it reach? How long does it take to fall? b) Express the ball’s height at 5 seconds: — Approximately, looking at the graph. — Using the analytic expression.

A closed geometric curve with two unequal perpendicular axes, resulting from Watch the video for target 15.c. Think of something you can do to contribute cutting the surface of a cone with a plane that is not perpendicular to its axis. Commitment to achieve that goal. Make a commitment to put your idea into practice. A procedure for solving a system of equations that involves solving for the same unknown in both equations and then equalising the resulting expressions.

equation

Practice makes perfect! In this section you will have to solve many different types of problems.

3 One of these equations is shown in the graph

9 10 time (h)

direct motion

306

Enterprising culture

• a) You are at a fountain and you have one 5-litre jar and one 3-litre jar. What would you do to measure exactly one litre of water?

Connect these nine dots with a line broken into four segments.

• Match each of these containers with its graph:

alternate exterior angle

axis of symmetry

Glossary. We learn the relevant terms that are underlined in the units with a clear definition.

•

— In the first two containers, the level rises uniformly. However, the level rises faster in the second container than in the first. — In the third container, the level rises slowly at first and then quickly.

Glossary absolute error

• Two rancher siblings share an inheritance equally. Nicole spends her part on buying a herd of 80 horses. Charlie spends his part on a herd of 100 cows. If a horse costs €150 more than a cow, how much was the inheritance?

When a tap is opened over a container, the height (h) of the liquid is a function of (depends on) the time passed (t). And when we graph this function, we see that each container has its own graph.

169

ment. By Self-assess s, se activitie doing the r u o heck y you can c ding and n ta unders you have how much learnt.

307

Academic and professional

ICT

orientation

Evaluation

Linguistic Plan

Learn how to obtain information, select it and apply it; to plan, manage and work on projects; to collaborate online in an ethical and safe manner.

Evaluate your personal skills, discover and awaken your calling, train yourself to make decisions and learn to choose between different options.

Discover different strategies to analyse what you have learnt and how you learnt it; train yourself to take responsibility or overcome difficulties.

Use your communication skills in the different types of text that you will see. Language is always present, communicate!

The resource bank is LIKE THIS www.anayaeducacion.es A space with resources, techniques and activities, designed to strengthen your knowledge. Access the resource bank by registering at anayaeducacion.es. You just need to have an email, the code indicated on the first page of this book and the permission of a parent or a legal guardian.

More about the keys

Resources related to

THE KEYS of the project SDG

SDG Commitment, SDG Commitment, with microvideos that will help you know what are the targets for reaching the Sustainable Development Goals worked on in this project.

Linguistic Plan, with infographics that will give you models to work with the four linguistic skills, using different types of texts (descriptive, narrative, explanatory…). Cooperative learning Preparing for the task In small groups, of four or five members: 1 All the members of the team will review how the assigned task can be accomplished. 2 To do this, the steps can be shared out to each team member who then in turn will explain how each part of the process can be done to the others. The others listen and participate if they think they can contribute something.

Authorship / adaptation : Variant of the Educational Innovation Laboratory of the colegio Ártica - David and Roger Johnson.

Developing thinking, where explanations are included on how to apply the different strategies for thinking proposed in the project.

3 Once everyone is in agreement on how to do each part, you will all complete the tasks and, finally, verify, among everyone, that you have solved it correctly.

Cooperative learning, which includes the description of the cooperative learning techniques proposed in the project. Emotional education, with resources to support you with overcoming worries generated in different situations in your learning process (beginning of the school year, taking a test...).

Thinking techniques Logic Wheel This thinking technique will help you to establish phases when analysing specific content that you have to study.

Identify What is it? What is it like? Are there different types?

By following a logical sequence (the logic wheel), and by asking yourself a series of questions in each phase, you can:

• Identify content by asking yourself: What is it? What is it like? Are there different types? • Compare the content by formulating questions such as: In what way is it similar to ...? In what way is it different from ...? • Establish cause-effect relationships by asking yourself questions such as: Why? What impact does it have ...? • Argue, assess and ask yourself questions such as: What conclusions can be drawn after the analysis? What can be assessed or scored about it? Doing a data dump of these questions into a graphic organiser will help you. Authorship: Hernández, P., and García, L. A.; adapted by Escamilla, A.

Compare

Argue, assess What can we conclude?

Logic wheel

3 Establish cause-effect relationships Why? What impact does it have ...?

In what way is it similar to ...? In what way is it different from ...?

ICT, by using data sheets that will reinforce your healthy, correct and safe use of information and communication technologies.

Academic and professional orientation, with information on different professions linked to the subject content.

Evaluation, which includes resources for your portfolio, as well as rubrics and targets that will facilitate your self-evaluation.

NOTEWORTHY

resources in the material Tutorials Activities with GeoGebra Glossary Learn by playing Self-assessments Language Bank

Resources classified

by unit

All the resources are classified so that you can easily locate the ones related with each section of your unit.

course contents FRACTIONS AND DECIMALS

Page 10

1. Fractions.................................................................................. 12 2. Operations with fractions................................................. 14 3. Decimal numbers................................................................. 16 4. Fractions and decimals with the calculator............... 20 Exercises and problems solved.......................................... 22 Exercises and problems......................................................... 23 Maths workshop........................................................................ 26 Self-assessment........................................................................ 27

POWERS AND ROOTS

Page 28

1. Exponentiation...................................................................... 2. Scientific notation................................................................ 3. Roots and radicals............................................................... 4. Rational and irrational numbers..................................... Exercises and problems solved........................................... Exercises and problems......................................................... Maths workshop......................................................................... Self-assessment.........................................................................

ARITHMETIC PROBLEMS

Page 44

1. Approximations and errors.............................................. 2. Calculating with percentages......................................... 3. Compound interest............................................................. 4. Common problems.............................................................. 5. Compound proportion in arithmetic problems........ Exercises and problems solved........................................... Exercises and problems.......................................................... Maths workshop......................................................................... Self-assessment.........................................................................

PROGRESSIONS

30 32 34 36 37 38 42 43

46 49 53 54 57 59 60 64 65

Page 66

1. Sequences.............................................................................. 2. Arithmetic progressions.................................................... 3. Geometric progressions.................................................... 4. Surprising geometric progressions............................... Exercises and problems solved.......................................... Exercises and problems......................................................... Maths workshop........................................................................ Self-assessment........................................................................

68 70 72 76 78 79 82 83

ALGEBRAIC LANGUAGE

Page 86

1. Algebraic expressions........................................................ 88 2. Monomials............................................................................... 89 3. Polynomials............................................................................ 90 4. Identities.................................................................................. 92 5. Dividing polynomials.......................................................... 94 6. Factorising polynomials.................................................... 97 7. Algebraic fractions.............................................................. 98 Exercises and problems solved........................................... 100 Exercises and problems.......................................................... 101 Maths workshop......................................................................... 106 Self-assessment......................................................................... 107

EQUATIONS

Page 108

1. Equations. Solving an equation...................................... 110 2. First-degree equations...................................................... 112 3. Second-degree equations................................................ 114 4. Polynomial equations with a degree greater than two.................................................................................. 118 5. Solving problems with equations.................................. 120 Exercises and problems solved........................................... 122 Exercises and problems.......................................................... 123 Maths workshop......................................................................... 128 Self-assessment......................................................................... 129

SYSTEMS OF EQUATIONS

Page 130

1. Lineal equations with two unknowns........................... 132 2. Systems of linear equations............................................. 133 3. Equivalent systems.............................................................. 134 4. Types of systems by number of solutions.................. 135 5. Methods for solving systems........................................... 136 6. Systems of non-linear equations................................... 140 7. Solving problems through systems.............................. 141 Exercises and problems solved........................................... 143 Exercises and problems.......................................................... 144 Maths workshop......................................................................... 148 Self-assessment......................................................................... 149

CHARACTERISTICS OF FUNCTIONS

Page 152

1. Functions and graphs........................................................ 154 2. Function aspects.................................................................. 156 3. Analytical expression of a function............................... 160 Exercises and problems solved........................................... 162 Exercises and problems.......................................................... 163 Maths workshop......................................................................... 168 Self-assessment......................................................................... 169

12 9

LINEAR AND QUADRATIC FUNCTIONS

Page 170

1. Proportionality function y = mx.................................... 172 2. Linear function y = mx + n.............................................. 174 3. Applying linear functions. Movement problems...... 177 4. Studying two linear functions together...................... 178 5. Parabolas and quadratic functions............................... 179 Exercises and problems solved........................................... 181 Exercises and problems......................................................... . 182 Maths workshop........................................................................ 186 Self-assessment........................................................................ . 187

METRIC PROBLEMS IN PLANE GEOMETRY

Page 190

1. Angle relationships.............................................................. 192 2. Similar triangles.................................................................... 194 3. Similar shapes. Scales........................................................ 196 4. Pythagorean theorem........................................................ 197 5. Algebraic applications of the Pythagorean theorem................................................................................... 198 6. Areas of polygons................................................................ 199 7. Areas of curved shapes..................................................... 200 8. Loci............................................................................................ 201 9. Conic sections as loci......................................................... 202 Exercises and problems solved.......................................... 204 Exercises and problems......................................................... 206 Maths workshop........................................................................ 212 Self-assessment........................................................................ 213

GEOMETRIC SHAPES

Page 214

1. Reguar and semiregular polyhedra.............................. 216 2. Truncating regular polyhedra.......................................... 218 3. Planes of symmetry of a shape...................................... 220 4. Axes of rotation of a shape.............................................. 221 5. Surface area of geometric shapes................................. 222 6. Volume of geometric shapes.......................................... 226 7. Geographic coordinates.................................................... 228 Exercises and problems solved........................................... 230 Exercises and problems.......................................................... 231 Maths workshop......................................................................... 236 Self-assessment......................................................................... 237

GEOMETRIC TRANSFORMATIONS

Page 238

1. Geometric transformations.............................................. 240 2. Motions on the plane.......................................................... 241 3. Translations............................................................................ 242 4. Rotations. Shapes with a centre of rotation.............. 244 5. Axial symmetries. Shapes with axes of symmetry.. 246 6. Composition of motions.................................................... 248 7. Mosaics, friezes and rosettes........................................... 250 Exercises and problems solved........................................... 252 Exercises and problems......................................................... 253 Maths workshop......................................................................... 258 Self-assessment......................................................................... 259

STATISTICAL TABLES AND GRAPHS

Page 262

1. The statistical process........................................................ 264 2. Statistical variables............................................................. 265 3. Population and sample...................................................... 266 4. Making a frequancy table................................................. 268 5. Using the right type of graph......................................... 270 Exercises and problems solved........................................... 272 Exercises and problems.......................................................... 273 Maths workshop......................................................................... 276 Self-assessment......................................................................... 277

STATISTICAL PARAMETERS

Page 278

1. Two types of statistical parameters.............................. 280 2. Calculating x– and q in frequency tables................. 282 3. Joint interpretation of x– and q.................................... 284 4. Position parameters: median and quartiles............... 286 – and q............................... 288 5. Using a calculator to get x 6. Statistics in the media........................................................ 289 Exercises and problems solved.......................................... 290 Exercises and problems......................................................... 291 Maths workshop......................................................................... 294 Self-assessment......................................................................... 295

CHANCE AND PROBABILITY

Page 296

1. Random events..................................................................... 298 2. Probability of an event....................................................... 300 3. Probability in regular experiments. Laplace’s law.......................................................................... 301 4. Probability in irregular experiments. Law of large numbers........................................................ 302 5. Probabilities in compound experiments..................... 303 Exercises and problems solved........................................... 305 Exercises and problems......................................................... 306 Maths workshop......................................................................... 310 Self-assessment......................................................................... 311

Annex • Glossary............................................................................................. 312

fractions and decimals Reading and listening

Sexagesimal fractions in Mesopotamia In ancient Mesopotamia, the Babylonians used the sexagesimal system to write numbers. Sexagesimal fractions have denominators that are powers of base 60. To express 2 they wrote 24 . 5 60 To express 1 they wrote 45 = 452 . 80 3 600 60

How the Babylonians wrote numbers To understand how the Babylonians wrote numbers on clay tablets, look at some examples in this table. It shows the place values of sexagesimal units: 602

1/60

1/602

→ 1 · 602 + 16 · 60 = 4 560 → 24 = 2 = 0.4 60 5 → 1 + 24 = 1.4 60 → 1 + 24 + 452 = 1.4125 60 60 Notice that this system only uses two symbols: = 10 and = 1 to write the numbers from 1 to 59. Depending on the position of those numbers, their values are multiplied by 1, 60, 602… or by 1/60, 1/602… This is what we call a positional system.

From sexagesimal fractions to decimal numbers To transform a number written in sexagesimal notation to a decimal, we just have to use what we already know. Observe: N = 1;24,45 (sexagesimal form) N = 1 + 24 + 452 = 1 + 2 + 1 = 5 80 60 60 N = 1 + 2 : 5 + 1 : 80 = 1.4125 (decimal form)

Finally, decimal fractions Integers have been written as decimals in the Western world since the 13th century. However, sexagesimal fractions were still used to write parts of the unit! Even today, we still use them to measure time: a minute is onesixtieth of an hour, and a second is one-sixtieth of a minute. Therefore, 3 h 36 min 45 s written in decimal time would be 3 + 36 + 452 = 3.6125 h. 60 60 Using decimals to write parts of the unit did not become popular until the end of the 16th century. The French François Viète and the Flemish Simon Stevin were the main driving forces behind the change. 1 In the text, you can find the expression ‘to be a driving force’. Discuss

its meaning with your classmates and write a sentence using it in which its meaning is clear.

2 True or false? Make the false ones right.

a) Sexagesimal fractions were used by the ancient Egyptians. They are fractions with denominators that are powers of base 62. b) The sexagesimal system only uses 6 symbols to write the numbers from 10 to 60. c) Integers started to be used in the Western world at the beginning of the 13th century. d) Nowadays, we keep using sexagesimal fractions to measure time. e) The mathematicians Viète and Steving had a big influence on the popularisation of decimal numbers at the beginning of the 16th century. Solve 1 Write 3 , 5 and 5 in the Egyptian style, as a sum of unit fractions.

4 6

2 The number 3;8,29,44 is in sexagesimal form. Write it as a sum of

fractions with denominators that are powers with a base of 60 (3 + + 8/60 + …) and convert it into decimal form. Do you recognise this number?

3 What numbers do you see in this

tablet? The colours of the columns correspond to the same units as the table on the previous page.

ANK ANK B E G A U LANG LANGUAGE B ANK ANK GE BANK B B E E G G A A U U GUA K LANG ANG N L A L AN GE BANK 11 ANK GE BANK B B E E G G A A U U G A LAN LANG LANGUA LANGU

FRACTIONS Integers are useful for counting things, but they are not as good for expressing measurements. To measure things, we often have to use parts of the unit, like one half, three quarters, seven thousandths... We write these measurements as fractions: 7 1 4 2 3 1000 A fraction is a quotient of two integers. This quotient can be an integer d 6 = 3, –12 = – 4n or a fractional number d 17 = 8 + 1 , –13 = –2 – 3 n. 2 3 2 2 5 5 If the numerator is a multiple of the denominator, the fraction is an integer. If not, it is a fractional number.

anayaeducacion.es • Activities to review operations with integers. • Activities to practise operations with integers.

Measuring with fractional numbers

The set of all the integers and all the fractional numbers is called the set of rational numbers and is represented by the letter Q. Rational numbers are numbers that can be written as a fraction.

To measure means to relate two magnitudes of the same type. When we say that the volume of the Moon is 1/50 the volume of the Earth, we are taking the volume of the Earth as a unit. And if we say that the visible part of an iceberg is 1/9 of its total, we are taking the entire volume of the iceberg as a unit.

Rational numbers can be placed on a number line. Between any two numbers on the number line there are infinite other rational numbers. 5 –— 2 –5

–4

–3

10 = 1 + — 3 — 7 7

1 –— 2 –2

–1

23 = 4 + — 3 — 5 5 3

ã Simplifying fractions Explain the name…

If the numerator and denominator of a fraction can be divided by the same number (other than 1 and –1), when we divide them by that number, we say that we have simplified or reduced the fraction. For example: 25 = 5 ; 8 = 4 = –2 ; 3000 = 30 = 2 15 3 –12 – 6 3 4500 45 3 When a fraction with a positive denominator cannot be reduced further, we say that it is irreducible.

Why do we use Q to name the set of rational numbers? It comes from the word quotient in English, because rational numbers are the quotient of two integers. Think and practise

1 Draw a number line like this one in your notebook and

put these numbers in their approximate place: 17 , – 11 , 20 , 2 , 16 , – 21 , – 7 3 4 5 3 7 5 2 –5 – 4 –3 –2 –1

2 , 2 , 5 , 10 , –20 , 30 , –30 , 40 4 6 10 15 30 40 –45 –60 5

4 Match each of the following fraction with its

corresponding irreducible fraction:

2 What fractions do these points on the number line

represent? A –3

B –2

–1

C 0

D 1

E 2

3 Simplify these fractions:

15 a) 6 b) 18 20

III) 3 4

III) –2 5

14 III) 1 IV) –3 c) –15 d) 8 –35 3 40

Unit

ã Equivalent fractions Cross-multiplication Cross-multiplication is a method that we use to check if two fractions are equivalent: a = c if a · d = c · b b d For example, 18 and 21 are 30 35 equivalent because: 18 · 35 = 630 = 21 · 30

anayaeducacion.es Activities to review simplifying fractions.

Every rational number can be written as many different equivalent fractions. In fact, each rational number has an infinite number of equivalent fractions: 3/5 = 6/10 = 9/15 = … We therefore need a way to recognise when two fractions represent the same rational number. Two fractions are equivalent when they can be simplified to the same irreducible fraction, which is the common expression of that rational number. 18 and 21 are equivalent because 18 = 18 : 6 = 3 and 21 = 21 : 7 = 3 . 30 35 35 35 : 7 5 30 30 : 6 5

ã Comparing fractions Two fractions with the same denominator are very easy to compare just by looking at their numerators. To compare two fractions with different denominators, we ‘reduce them to a common denominator’. In other words, we find two fractions with the same denominator that are equivalent to the ones we have.

Problems solved 1 Indicate

whether these fractions are equivalent: a) 9 and 12 39 52 b) 15 and 38 35 57 8

9 = 9 : 3 = 3 ; 12 = 12 : 4 = 3 → They are equivalent. 39 39 : 3 13 52 52 : 4 13 b) We do cross-multiplication and check whether they are equivalent: 15 · 57 = 855; 38 · 35 = 1 330 → They are not equivalent.

2 Compare 7 , 5 and 9 .

a) We find their irreducible fractions and check whether they are equivalent:

We take the LCM as the denominator. LCM (12, 8, 16) = 48. 48 : 12 = 4 → 7 = 7 · 4 = 28 12 12 · 4 48 48 : 8 = 6 → 5 = 5 · 6 = 30 8 8 · 6 48 48 : 16 = 3 → 9 = 9 · 3 = 27 16 16 · 3 48

Clearly: 27 < 48 Therefore: 9 < 16

28 < 30 48 48 7 <5 12 8

Think and practise 5 True or false?

a) 2 > – 7 b ecause the first one is positive and the 5 4 second one is negative. b) 7 > 2 b ecause the first one is greater than 1 and 3 5 the second one is less than 1. c) 8 > 7 b ecause the first one is greater than 2 and 3 4 the second one is less than 2. d) – 8 > – 7 b ecause the first one is greater than –2 3 4 and the second one is less than –2.

6 Use simplification and cross-multiplication to figure

out whether or not these fractions are equivalent: 36 and 78 a) 12 and 21 b) 20 35 102 221

7 Find equivalent fractions of 60 …

a) … with numerator of 20.

126

b) … with denominator of 42. 8 Order these fractions from smallest to largest:

7 12

–6 4

4 6

– 3 15

5 9

–1 2

3 4

13 18

OPERATIONS WITH FRACTIONS ã Adding and subtracting fractions To add (or subtract) fractions with the same denominator, we add (or subtract) their numerators and keep the same denominator.

Mental arithmetic a) 2 + 5 – 4 3 3 3 c) 1 + 1 2 4 e) 17 – 3 5

To add (or subtract) fractions with different denominators, we first transform them into equivalent fractions with the same denominator, then we add (or subtract) their numerators.

b) 1 – 2 3 d) 7 – 1 5 f ) 17 – 5 3

For example:

7 – 5 + 2 = 42 – 25 + 120 = 42 – 25 + 120 = 137 10 12 60 60 60 60 60

ã Multiplying and dividing fractions The product of two fractions is a fraction in which the numerator is the product of the numerators and the denominator is the product of the denominators: a · c = a ·c b d b·d For example: 8 · 7 = 8 · 7 = 56 = 28 3 10 3 · 10 30 15 The quotient of two fractions is the product of the first fraction and the inverse of the second fraction: a : c = a · d = a·d b d b c b ·c 6 :3= 6 · 1 = 6 = 2 For example: 9 : 5 = 9 · 7 = 63 ; 11 11 3 33 11 4 7 4 5 20

anayaeducacion.es • Activities to review adding and subtracting fractions. • Activities to practise adding and subtracting fractions.

Mental arithmetic 4 · 15 a) 3 · 7 b) 9 5 8 c) 1 · 12 d) 1 · 2 · 3 2 13 2 3 5 e) 6 : 3 f ) 6 : 6 5 5 5 g) 6 : 1 h) 1 : 1 3 6 5 2

ã Combined operations with fractions To do combined operations with fractions, we first solve the expressions in brackets and then the rest of the operations. Remember to solve products and quotients before additions and subtractions. For example: 2 – 1 c1 – 1 m + 2 (–2) = 2 – 1 · 3 + 2 (–2) = 2 – 1 – 4 = 3 3 3 4 3 4 4 3 = 24 – 3 – 16 = 5 12 12 12 12

Think and practise

anayaeducacion.es Activities to practise combined operations with fractions.

Calculate and simplify the results: 1 a) 7 + 11

c) 3 · 4 5 f ) 4 : 1 5 6

b) 6 – 11 4 4 : 6 d) 6 : 4 e) 5 5 9

d 13 – 7 n · d 9 + –13 n 2 a) d 3 + 7 – 7 n : 25 b)

1 – d 3 – 1n (–3) · d 3 – 5 2 4 3 a) b) 3 +1 (–2) · d 4 – 4 3

1n 3 6n 5

d 2 – 5 n·d 3 – 5 n 3 – 1 ·d 3 – 2 n 5 4 15 3 9 4 6 4 a) b) d 7 – 5 n · 4 +1 6 + 4 ·d 1 – 3 n 25 2 4 12 6 3

Unit

ã Fraction of an amount Mental arithmetic 1 Find: a) 1 of €520 000. 2 b) 3 of 1 000 000 people. 5 c) 7 of 500 buildings. 10 2 Find the total amount: a) 1 of the total is 350. 2 b) 2 of the total is 400. 3 c) 7 of the total is 350. 10

F ocu s on Eng lish pay off: to give someone all the money you owe them.

Observe All the portions (fractions) of a whole add up to 1. For example: We divide a cake by giving Ana 1/3, Mark 1/4 and Oliver the rest. How much does Oliver get? 1 – c 1 + 1 m =1 – 7 = 5 12 12 3 4

1 What fraction completes the unit? a) 1 , 2 c) 1 , 4

1 and ? b) 2 , 4 ? 3 1 and ? d) 1 , 6 ? 2

1 and ? 6 ? 1 , 1 and ? 4 8 ?

To find 3 of an amount, like for example of €1 200, we divide it by 5 (to get one 5 fifth) and then, we multiply it by 3. In other words, we multiply the amount by 3 → 3 · €1 200 = €720 5 5 To find a fraction, a , of an amount, C, we multiply a · C. b b Examples

• A postman delivers 3/28 of a total of 4 004 letters. How many does he deliver? 3 of 4 004 = 3 · 4 004 = 3 · 4 004 = 3 · 143 = 429 letters 28 28 28 • Bertha owns 7/20 of a company. This year, she received €37 800 in profits. What are the company’s total profits? If she receives €37 800 for owning 7 , then 1 is 37 800 = €5 400. 20 20 7 Therefore, the total d 20 n is 20 · 5 400 = €108 000. 20 We can get the same result by multiplying Bertha’s share of the profits (€37 800) by the inverse of her share of the company, 20 . 7 7 of the total = 37 800 → total = 37 800 · 20 = €108 000 7 20 To find a fraction, a , of another fraction, c , of an amount, C, we multiply b d a · c ·C . b d Example

Three siblings receive an inheritance of €104 000. Albert gets 3/8, Bertha gets 5/12 and Clare gets the rest. Clare uses 2/5 of her portion to pay off * her debts. How much does she have left? 1 – 3 – 5 = 24 – 9 – 10 = 5 is Clare’s portion. 8 12 24 24 Since she spends 2 of her portion, she has 3 left: 5 5 She has 3 · 5 · 104 000 = 1 · 104 000 = €13 000 left. 8 5 24

Think and practise 5 A cyclist has completed 5/9 of today’s 216 km stage.

How many kilometres has he gone?

6 Yesterday, Karen decided to take €3 900 out of the

bank, which is 3/11 of her savings. What are her total savings?

7 A pool with 5 250 litres of water will be used 4/15 by

Barbara, 2/5 by Eric and the rest by Roberta. Roberta uses 3/10 of her portion to water tomatoes and the rest to water her fruit trees. How much water does Roberta use to water her fruit trees?

DECIMAL NUMBERS Decimal numbers are often used for measurements because they can express any intermediate value between two integers. Decimal numbers can be represented on the number line, and we can use them to get as close as we want to any number on the line:

Remember On calculators, like in English, we use a full stop, not a comma, as the decimal point. 1 437.54 → {∫∫‘¢«|…∞¢}

–6

–5

–4

–3

–2

–1

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.8 anayaeducacion.es Decimal numbers on the number line.

3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89

3.9

The red point can be written as a decimal as close to the point as we want (3.857…). Writing numbers in decimal form gives us a very easy and effective way to assess them, compare them and operate with them.

ã Types of decimal numbers Let’s look at the different types of decimal numbers: • Terminating decimals have a limited number of decimal places. For example: 5.4; 0.97; 8; –0.0725 • Recurring decimals have an infinite number of decimal places that repeat periodically.

Remember In a number, the group of decimal places that is repeated again and again is called period. We draw an arc over repeating figures to indicate the period:

# 5. 68

! 16.147

— In pure recurring decimals, all the figures after the decimal point are # repeated. For example: 7.81818181... = 7. 81 — In mixed recurring decimals, at least one of the figures after the decimal ! point is not repeated. For example: 18.35222222... = 18.352 • Decimals that are not terminating or recurring have an infinite number of figures that do not repeat regularly. Unlike terminating decimals and recurring decimals, these numbers are not rational. They are called irrational numbers. For example: 2 = 1.4142135…; π = 3.14159265…

Think and practise 1 What type of decimal are these?

3.52 2.7

! 2.8

3.5222…

# 1. 54

2 Order these numbers from smallest to largest:

3 = 1.7320508… π – 2 = 1.1415926…

! 2.5

2.5

! 2.35

2.505005… ! 3 Write three numbers between 2.5 and 2.5 .

Unit

ã Transforming fractions into decimals To write a fraction in decimal form, we divide the numerator by the denominator. The quotient can be: • An integer, when the numerator is a multiple of the denominator. For example: 72 = 8; –240 = –16 15 9 anayaeducacion.es Help with reasoning: transforming fractions into terminating and recurring decimals.

• A terminating decimal, if the only prime factors of the denominator of the simplified fraction are 2 and 5 (or either of them). For example: 3 = 0.375; 123 = 3.075; 42 = 1.68 25 8 40 Look at why this is true: 123 = 123 = 123 · 5 2 = 123 · 25 = 3 075 = 3.075 40 2 3 · 5 2 3 · 5 3 1000 10 3 If the only factors are 2 and 5, we can always write the denominator as a power of base 10.

Example

it repeats

7 3.0 20 0.428571 60 40 50 10 The quotients and remainders repeat 3 from here on.

• A recurring decimal, if the denominator of the simplified fraction has a prime factor other than 2 or 5. ! # # For example: 11 = 3.6 ; 86 = 7.81 ; 87 = 29 = 1.318 3 11 66 22 If the quotient is not a terminating decimal, why can we be sure that it is recurring? Let’s look at an example: 3 : 7. You can see the division on the left. When we divide by 7, the remainder can only be 1, 2, 3, 4, 5 or 6, so it will have to repeat at some point. After that, the entire sequence will repeat. All irreducible fractions can be written in decimal form: • Terminating decimal, if the only prime factors of the denominator are 2

and 5.

• Recurring decimal, if the denominator has prime factors other than 2

and 5.

Therefore, they are both rational numbers. However, decimals with infinite decimal places that do not repeat are irrational numbers. Think and practise 4 True or false?

! a) 1 = 0.333… = 0.3 3 ! 3 = 3 · 0.333… = 0.999… = 0.9 3 ! Because 3 = 1, then 0.9 = 1. 3 ! # b) 5.4 = 5.44 # # c) 3.72 = 3.7272727… = 3.727 ! ! d) 0.3 + 0.6 = 1

5 Without doing the division and looking only at the

denominator of the simplified fraction, say whether these fractions transform into terminating decimals or into recurring decimals: 101 d) 1001 42 c) a) 44 b) 1024 500 150 150 6 Write a value of k that makes the fraction 84 : k a) An integer. b) A terminating decimal. c) A recurring decimal.

3 DECIMAL NUMBERS

ã Transforming decimals into fractions We have just seen that if we divide the numerator of a fraction by its denominator, the result is a terminating or recurring (pure or mixed) decimal number. Now, let’s look at the inverse case: How do we transform a decimal into a fraction? ➜ from terminating decimals to fractions

Writing a terminating decimal as a fraction is very easy because the denominator is a power of base 10. For example: 2.5 = 25 = 5 ; 3.41 = 341 ; 0.004 = 4 = 1 1000 250 100 10 2 ➜ from pure recurring decimals to fractions

Let’s look at two examples of the process: ! • Period with one figure: N = 5.4 = 5.4444… When we multiply N by 10, the result is another number with the same decimal portion.

When we multiply N by 1 000, the result is another number with the same decimal portion.

10N = 54.444… 3 The decimal portion disappears when we subtract: 10N = 5.444…

10N – N = 54 – 5 → 9N = 49 → N = 49 9 & • Period with more than one figure: N = 6.207 = 6.207207207…

1000N = 6 207.207207… The decimal portion disappears when we 3 1000N = 6.207207… subtract:

1 000N – N = 6 207 – 6 → 999N = 6 201 → N = 6 201 999 You can check these two examples by doing the divisions on a calculator. anayaeducacion.es Help with reasoning: transforming from pure recurring decimals into fractions.

To write a pure recurring decimal, N, as a fraction: • We multiply N by a power of base 10 to find another number with the

same decimal portion.

• When we subtract them, the result is an integer. • We isolate N to get our answer.

Think and practise 7 Write as fractions:

8 Notice that 0.208 + 0.791 = 0.999 = 1.

a) 6.2 b) 0.63 c) 1.0004 Check by writing each addend as a fraction and then adding the fractions together. ! ! ! 0.1 f ) 2.7 d) 3.5 e) 9 Transform these decimals into fractions and then & & # g) 0.23 h) 41.041 i) 40.028 calculate: ! ! ! ! & & # # 1.3 : 2.16 a) 3.5 + 1.76 – 2.103 b) j) 5.9 k) 7.009 l) 0.99

Unit

➜ from mixed recurring decimals to fractions

• To transform N = 2.563 into a fraction:

N = 2.5636363… We multiply by 10 to get a pure recurring decimal.

10N = 25.636363… Now, we multiply by 100 to get another number with the same decimal part. 1 000N = 2 563.636363… When we subtract this number from the previous number, the decimal part disappears. In other words, the result is an integer.

1 000N – 10N = 2 563 – 25 → 990N = 2 538 → N = 2 538 990 & • Another example: N = 0.07324 = 0.07324324324… 100N = 7.324324… We get a pure recurring decimal. anayaeducacion.es Help with reasoning: transforming mixed recurring decimals into fractions.

100 000N = 7 324.324324… Another pure recurring decimal, with the same decimal portion. 100 000N – 100N = 7 324 – 7 → 99 900N = 7 317 → N = 7 317 99 900 Check both cases with a calculator. To write a mixed recurring decimal, N, as a fraction:

anayaeducacion.es GeoGebra. Examples of how to write decimal numbers as fractions.

• We multiply N twice by powers of base 10 to get two pure recurring

decimals with the same period.

• When we subtract them, the result is an integer. • We isolate N to get the fraction. ➜ summary

To write a recurring decimal (pure or mixed) as a fraction, we use the given number to find two pure recurring decimals with the same period. When we subtract them, the result is an integer. We already know that decimal numbers with infinite non-repeating decimal places are irrational numbers, so they cannot be written as fractions. Think and practise 10 Complete the process to write these numbers as

fractions:

N = 6.21777… ! a) 6.217 * 100N = 621.77777… 1000N = 6 217.7777… N = 0.0316262… # b) 0.03162 * 1000N = 31.626262… 100 000N = 3162.626262…

11 Write these decimals as fractions:

! ! # 0.001 c) a) 6.25 b) 5.018

12 Which of these numbers are rational? Write them as

a fraction:

# c) 5. 03 & d) 0.3212121… e) π = 3.141592… f ) 7.4331 # # 13 Transform into fractions and check that 5.48 = 5.484 . a) 3.51

b) 5.202002000…

4 Secondary functions On scientific calculators, most keys have two secondary functions (they appear above the key). The two functions are usually in different colours: • SHIFT → yellow • ALPHA → red For example, this key:

When you press it calculates the cube root. When you press you can write recurring decimals. We will learn more about this on the next page. From now on, when we speak about a secondary function, we will refer to it with its key. For example, to speak about the cube root function we will write:

FRACTIONS AND DECIMALS WITH THE CALCULATOR This school year is a good time to start working with a scientific calculator, which will be very useful to you during the rest of Secondary and Bachillerato. We will be referring to the CASIO CLASSWIZ calculator in many of our instructions because it is the most widely used by students of this level. But you could use any other calculator with similar characteristics.

ã Configuration We mostly use the calculator to do arithmetic calculations. To do so, enter the � and choose 1:Calculate. It is essential to configure the calculator to receive data (INPUT) and express results (OUTPUT) in the format that we need. We suggest configuring both in mathematics mode. This way, all fractions, roots and powers will be displayed in the way we are used to seeing them.

input

output

To do this, press the configuration key �. This select 1:Input/Output and then, 1:I Mat/O Mat (INPUT and OUTPUT in mathematics mode). It is also important to configure the calculator to OUTPUT in fractions and not in mixed numbers. To do this, go into the configuration menu ( �) and use the ’ arrow to move to the next screen. Once you are there, select 1:Fraction result. Then, select 2:d/c.

ã Fractions key and the ”’‘“ arrows.

To enter fractions, use the Enter 3 → 4 Recall what mixed numbers are The sum 3 + 2 can be written as: 5 3 2 . This is called a mixed number. 5 They are not used often nowadays, so we are not going to spend more time on them.

3 3 ’ 4 ” 4

3 If you press = 4

3 4

If you enter a fraction that is not simplified, you can press = to simplify it: 6 → 8

6 6 ’ 8 ” 8

6 = 8

3 4

Think and practise 1 Enter the expressions on the right into a calculator and

check that when you press the = key, the fractions are simplified or you get the corresponding fractions.

27 8 c) a) 3 b) 12 5 15 d) 3.25 e) 0.27 f ) 0.321

Unit

ã Operating with fractions Some simplifications You can try different methods on the calculator to find new, and sometimes easier, ways of doing things. For example, to enter 6 you can press:

6 8= 8

3 4

To do operations with fractions, just enter the chain of operations in INPUT and press the = key. For example, to get 2 + 5 · 1 – 11 : 3 6 12 2 ’ 3 ”+ 5 *

1 ’ 6 ”-

2 1 11 + 5 x – 11 ’ 12 ”= 3 6 12

7 12

If you make a mistake when entering data, you can always use the � key to go back. In other words, the � key deletes what is to the left of the cursor. If you want to go back to INPUT once you have finished entering data, press the “ arrow. You will be able to enter additional addends or correct any errors.

ã Decimals Non-recurring decimals are written in the usual way, remembering to use a full stop (.), not a comma, for the decimal point.

Be careful! If you enter a terminating decimal in which one or more figures repeat ‘many’ times, the calculator will probably interpret it as a recurring decimal: 5.43434343434343 =�

5.43434343434343 5.43

If you press � with a number in the OUTPUT, it will transform the number from a fraction into a decimal or vice versa. 3.875 → 3 . 875 = Use the

3.875 31 8

�

3.875 3.875

keys to enter a recurring decimal.

# 5.491 → 5 . 4

91 =

5.491 5437 990

�

5.491 5.491

Calculators have limits on entering recurring decimals and fractions due to the excessive size of the INPUT or OUTPUT. You can explore and find these limits on your own. Think and practise 2 Use the calculator to find the fractions that generate

these decimal numbers: # # # a) 2.354 b) 3.002 c) 0.0243 d) 3.701 ! ! # # 2.09 g) 1.1 e) 0.125 f ) 0.1233 h)

3 Use a calculator to solve the following. Write the

result in fractional and decimal form. d 4 + 1n : 2 5 5 d 4 – 5 n· 7 9 3 8 –1 3

4 Use a calculator to do these operations with fractions

and decimal numbers. Get the results as fractions and decimal numbers (terminating or recurring). a) 5 – 2 4 7

d 4 + 2n · –3 b) 9 5

d –2 – 3 n – 2 d) 5 7 & e) 2 – d 1 + 3n : 1 f ) 0.218 : d2 – 5 n 7 8 3 3 ! ! g) –5 – 3.25 h) d 2 – 3. 3n · 1 7 8 2

c) d–3 + 1 n : 2 3 5

BLEMS O R P d n a s e is exerc

solved

1 Operations with fractions

Calculate and simplify: 1+

1 1+

1+ 1 2

1 1+

c1 + 1 m 2

=1+

1+ 1 2 +1 2

=1+

1 + c1 : 3 m 2

=1+

1+ 2 3

= 1 + c1 : 5 m = 1 + 3 = 8 3 5 5 1 Your turn Calculate: 3+ 5 3+ 1 2

2 Operations with fractions and decimals

Calculate and give your answer as a fraction: ! 1 + 0.! 3 – 3 · 1.02 + 1.5 5 2 Your turn Calculate: ! ! ! ! (4.28 – 0.12) 0. 3 : 1. 6 + 7 · 0.4 3

We transform all the decimals into fractions and complete the calculations. ! 0. 3 = 1 ; 1.5 = 3 3 2 ! ! ! N = 1.02 → 10N = 10.2 → 100N = 102.2 ! ! 100N – 10N = 102.2 – 10.2 = 92 → 90N = 92 → N = 92 = 46 90 45 Now, we calculate:

1 + 1 – 3 · 46 + 3 = 1 + 1 – 23 + 3 = 6 + 10 – 46 + 45 = 15 = 1 5 3 2 45 2 5 3 15 2 30 30 30 30 30 2 3 Calculate the total

Annabelle spends 2/3 of her monthly pocket money in the first half of the month. Of what remains, she spends 3/5 in the second half, and has €10 left to save. How much is her monthly pocket money? Your turn We take half the oil from a bottle, and then one fifth of the oil that remains. If there are 3 L in the bottle now, what is its capacity?

If she spends 2 , she has 1 left over. 3 3 We calculate 3 of 1 → 3 · 1 = 1 5 3 5 5 3 At the end of the month she has:

1 – c 2 + 1 m = 15 – 10 – 3 = 2 3 5 15 15 15 15

If 2 of her monthly pocket money is €10, her monthly pocket money is: 15 10 · 15 = 150 = €75 2 2

4 Taps and fractions

Tap A fills a water tank in 2 hours, and tap B fills the same tank in 3 hours. The tank has a drain that takes 6 hours to empty the tank with the taps closed. If we open both taps and the drain, how long will it take to fill the tank? 22

If tap A fills the tank in 2 hours, it will fill 1/2 of the tank in one hour. Tap B fills 1/3 of the tank in one hour. The drain empties 1/6 of the tank in one hour. If we open all three at the same time, in 1 hour they will fill: 1/2 + 1/3 – 1/6 = 2/3 of the tank So the time they will take is: 1 : 2 = 3 h = 1.5 h = 1 h 30 min. 3 2

you ‘Portfolio’, urce bank on.es reso ci ca lio du fo ae port In the anay create your on how to guidance

will find

Unit

roblems

p Exercises and

Practise Fractions and decimals

Simplify these fractions and group the ones that are equivalent: 24 36

26 65

225 400

66 165

3 – 1 6 8

5 – 5 12 3

Order from smallest to largest: ! ! # a) 3.56; 3.56 ; 3.5 ; 3.56 ! ! # b) –1.32; –1.32 ; –1.32 ; –1.3 ! c) 2. 3; 8 ; 2.34; 32 ; 21 3 15 10

Write as fractions: ! ! ! 0.32 a) –1.03 b) 14.3 c) – 2.5 d) ! ! # # 5.345 h) 9.09 e) 0.012 f ) – 3.15 g)

Problem solved

Calculate the value of x that makes these fractions equivalent: a) x and 26 b) 3 and 51 x 17 4 6 a) The fractions are equivalent if their crossed products coincide: x · 4 = 26 · 6 → 4x = 156 → x = 39 b) We use cross-multiplication again: 3 · 17 = 51 · x → 51 = 51x → x = 1 4

343 539

Reduce to a common denominator and order from smallest to largest: 11 – 7 24 4

26 39

Operating with fractions

• 8 = 6 + 2 = 6 + 2 = 2 + 2 3 3 3 3 3 a) 8 b) 15 c) 16 5 7 8 6

e) – 7 3

Write each of these fractions as a decimal number: 5 233 13 17 9 13 23 25 9 6 200 7 990 22

4 5

13 9

7 · 11 3 · 52

19 22 · 5

3 · 7 2 · 23 5· 7

Classify these rational numbers as terminating decimals or recurring decimals (try to answer before doing the division): 13 17 4 2 1 81 3 5 50 11 60 250

Reduce to a fraction: 7·3 1–2 3+ 1 8 5 3 4 2 a) b) c) 1–1 5– 7 7– 3 5 2 2 6 12

Calculate and simplify by factorising, like in the example: • 15 · 7 = 15 · 7 = 3 · 5 · 7 = 1 21 25 21 · 25 3 · 7 · 5 · 5 5

Without doing the division, determine which of these are terminating decimals and which are recurring decimals: 3 2

d) – 3 2

Mental arithmetic: a) Half of 2 . b) One third of 12 . 3 7 c) Two-thirds of a number is 22. What is the number? d) Five-fourths of a number is 35. What is the number?

Find the value of x :

Write as a sum of an integer and a fraction like in the example:

Mental arithmetic: 2 · 9 a) 1 + 1 b) 1 – 1 c) 3 5 2 4 3 4 d) 2 of 60 e) 12 : 3 f ) 8 · 5 15 3 7

32 = 12 a) x = 35 b) 18 42 x 15 5

Write three numbers between each pair of decimals: ! a) 0.345 and 0.346 b) 2.3 and 2.4 c) – 4.5 and – 4.4

6 · 5 c) 12 · 35 a) 3 · 20 b) 5 21 25 18 7 36 90 · 14 13 · 84 f ) d) 9 · 20 e) 12 65 16 27 35 36 16

Transform into fractions and calculate: ! ! 0.12 – 0.2 a) 3.5 + 2.3 b) # ! ! ! 3.42 + 7.6 c) 1.6 – 1.02 d) Check that your answers, in order, are –7/90, 35/6, 122/11 and 29/45. 23

Exercises and problems 17

Calculate and give each result as an irreducible fraction: a) 3 – 1 d–1 + 2 n – 7 : d 4 – 1 + 2 n 5 2 3 15 5 3 b) d1 + 1 n – d 3 + 1 n · d 1 – 1 n : 1 3 4 2 3 4 6

One barrel of wine fills 480 bottles that measure 2/5 litre each. How many 3/4 litre bottles can be filled with the same barrel of wine?

The nutritional information written on the bottles of a brand of milk states that there are 120 mg of calcium in every 100 mL of milk. That is 3/20 of the recommended daily amount of calcium that a person should take every day. What is the recommended daily amount of calcium?

c) d 3 + 1 n – >1 – d 3 – 1 n + 2 – 3 H 5 3 4 2 3 20 d) – 4 · 1 + 3 – d 1 + 1 : 2 n 3 2 3 3 2 4 e) d 5 – 5 + 2 · 1 n : >2 – 1 d1 + 5 nH 2 2 6 3 4 3 f ) 5 : d 2 + 1n – 3 : d 1 – 1 n 4 2 4 g) – 3 >3 – 3 – d 17 – 1n · d 1 – 3nH 3 8 5 20

3/5 of a theatre’s seats are stalls, 1/4 are in the first balcony and the remaining 90 are in the second balcony. How many seats does the theatre have in total?

The solutions, in order, are: 11/4, –7/30, –1, –26/3, 17/24, –3/4 and –1/3. 18

If m = 1 and n = –7 , calculate: 3 2 4m + 1 n 2 b) 1 a) 1+ m m·n – 1– 1 n m– 1 n

True or false? ! a) 4 – (0.75 + 0. 6) + 13 = 1 3 12 ! ! b) d 5 + 0.16nd– 4 n + 65 d0. 1 – 0.2 – 1 n = 17 6 3 8 3 36 ! ! 1 : 3 – 1. 3 : 1.1 3 4 80 c) ! 2 ! = – 51 15 · 0.02 + – 1.09 3

Problem solving 20

A tank holds 1 500 L of water. We use 5/12 of the water one day and 500 L of water the next day. What fraction of the total water is left in the tank?

Julie receives €120 for her birthday. If she spends 2/5 on clothes, 1/4 on books and 3/20 on food, how much did she spend on each? What fraction of the money does she have left?

Of the 28 students in a class, 4/7 passed all their subjects. Of them, 1/4 got ‘excellent’ as their average score. How many students got ‘excellent’ as their average score? What portion of the class failed at least one subject?

Anne used some of her money to buy some comics. All the comics that she bought cost the same. If she used one fifth of her money to pay for one third of the comics that she bought, what fraction of her money was left after paying for all the comics?

We sold 2/3 of a property and then we sold 3/5 of the rest. The remaining 600 m2 will be used to make paths and gardens. What is the total area of the property?

One third of the people who attend a conference are from Spain and 3/10 are from France. Of the remaining attendees, 6/11 are from Switzerland and 25 people are from Italy. How many people attended the conference?

Michael spends 3/5 of his monthly pocket money in the first 2/3 of the month. If he spends at the same rate during the rest of the month, what fraction of his monthly pocket money will be left at the end of the month?

Two boxes of apples sell for €2.50 per kilo. The first box has 5/12 of the total and sells for €50. How many kilos of apples were there in each box?

Unit

Advanced problem solving

Let’s think!

Estimate to complete these equalities in your notebook with the missing figures:

There is a general rule for writing recurring decimals as fractions. It uses a different method from the one you learnt in this unit:

84 = 44 = 4 a) 436 = 44 b) 315 45 45 2156 77 75 = 45 = 4 c) 343 = 4 d) 534 11 445 84 27 32

decimal without the decimal point

Fill in the gaps in your notebook with the signs +, –, · or : to make each equality true:

Two farmers, a father and his daughter, take 2 hours to plough a field. The father takes 6 hours to do the same work alone. How long would it take the daughter to do it alone?

Two 600-millilitre bottles have orange juice in them. One bottle is one-third full and the other is two-fifths full. We add water to each bottle to fill them completely. Then, we empty them both into a larger bottle. What fraction of the liquid in the larger bottle is orange juice?

At a party, 2/3 of the guests are boys, 3/5 of the girls have a partner and 6 of the girls are single. How many guests were at the party?

I spend 1/10 of the money in my piggy bank. Then, I deposit 1/15 of what I have left in the bank. I still need €36 to have the original amount again. What was the original amount? A group of friends go to a pizzeria and order three kinds of pizza: A, B and C. Each of them eats 1/2 of A, 1/3 of B and 1/4 of C. They order a total of 17 pizzas and there are no whole pizzas left over. a) Did each friend eat more or less than a whole pizza? How many friends are they? b) How many pizzas of each kind did they order? Were there leftovers? c) Answer the same questions if the friends ordered a total of 20 pizzas.

non-recurring portion of the number without the decimal point as many zeros as there are figures before the period

Check that it works for these recurring numbers: & & ! ! a) 11.123 b) 0.7 c) 3.2501 d) 0.02171 Find four fractions between 1 and 1 . How 12 11 many are there?

A tap fills a water tank in 9 hours. If the tap and drain are both open at the same time, it takes 36 hours to fill the tank. How long does the drain take to empty the tank if the tap is closed?

as many nines as there are figures in the period

–

# # 3.27 = 327 – 3 18.2573 = 182 573 – 1825 99 9900

4 4 10 4 1 = 4 a) 2 4 2 4 3 = 17 b) 5 3 2 3 5 3 10 30 33

True or false? Explain and give examples.

a) Some decimal numbers are not rational. b) The quotient of two terminating decimals is always a terminating decimal. c) When you add up two pure recurring decimals, you always get a pure recurring decimal. d) All integers can be written as fractions. e) If two positive fractions are less than 1, their product can be greater than 1. f ) When you divide two recurring decimals, you always get a recurring decimal. 42

Divide the numbers from 1 to 10 by 11.

a) How many different decimals can you get? b) Is that related to the fact that we are dividing by 11? c) Can you predict the result of 23 : 11 and 40 : 11? 43

If we write the fraction 20/13 as a decimal number, what number is in the 50th position? Is the same number in the 100th position? If 0 < a < c < 1, which of these statements is b d true?

a · c > c c) a · c >1 a) a · c < a < 1 b) b d b b d d b d 25

Maths worksh

LET’S LEARN AND CALCULATE Identification codes and check digits Nowadays, codes are used to uniquely identify products, they are similar to an ID card for people or a number plate for cars. Many of these codes have a digit that can detect any errors made when writing them. These are called check digits. For example:

Bar codes You have certainly seen many bar codes. The bars and white spaces form a code in a binary system, which is a system that only contains ones and zeros. An optical device is used to read the code and identify the item. There are different types of bar codes, but the most common one has 13 figures grouped into three parts. You can see an example on the right. Let’s see how the check digit is calculated:

Country code (2 or 3 digits) 84 → Spain

1. We add up the figures in the odd positions, starting from the left (the check digit, x, is written as an addend to the sum). 2. The sum of the digits in the even positions multiplied by 3 is added to the previous result. We give x a value to make the total result a multiple of 10. Let’s look at an example of a bar code for a book like the one you are reading now. The 978 at the beginning represents the ISBN (International Standard Book Number). 978846785212x We add up the figures in the odd positions: 9 + 8 + 4 + 7 + 5 + 1 + x = 34 + x We add up the figures in the even positions: 7 + 8 + 6 + 8 + 2 + 2 = 33 We multiply the previous result by 3: 3 · 33 = 99 The total is: 34 + x + 99 = 133 + x. It has to be a multiple of 10. The only valid single-figure number for x is 7, making the sum 140. The number would therefore be: 9 7 8 8 4 6 7 8 5 2 1 2 7 • Calculate the missing control digits in the bar codes on the right. •

Copy the numbers of three bar codes from any three products in your notebook. Check that they are all correct by calculating the control digit.

INDEPENDENT PROBLEM SOLVING Commas matter If we put the comma in the correct place, this statement is true: ‘five times four twenty plus one, twenty-two’ Can you clarify it? 26

Code indicating the company and the product (9 or 10 digits)

Check code (1 digit)

84 3448504835 6

Unit

PRACTICE MAKES PERFECT! • A jeweller gets a €140 discount on 16 identical brooches. According to the catalogue, their price is €87.50 each.

At what price should he sell each brooch if he wants to make €500 in profit? • Marta buys three biscuits and Beatrice buys two. Their friend Veronica joins them later for breakfast but she does not bring any biscuits. When they divide up the cost, Veronica has to pay €5. How will Marta and Beatrice divide up the €5?

• A group of friends goes to a coffee shop. They all order coffee and one-fifth of them also order cake. A coffee costs €0.85 and a piece of cake costs €1.10. They give the waiter €11. Did they leave a tip? If so, how much of a tip did they leave? • A landowner hires a servant for an annual salary of eleven gold coins and one horse. After four months, the servant quits and receives the horse and one coin. What was the value of the horse?

SELF-ASSESSMENT

anayaeducacion.es Answer key.

1 Calculate and simplify the result:

7 I can fill twenty 3/5-litre bottles of oil with

1 + 5 · c 5 + 1m – 1 : 2 8 3 3 5 2 Write a fraction that transforms into a terminating

decimal, a fraction that transforms into a pure recurring decimal and a fraction that transforms into a mixed recurring decimal.

3 Write three numbers between the pairs of numbers

given:

! ! 2.7 and 2.8 a) 3 and 4 b) 20 25 4 Without doing the division, say whether these are

terminating decimals or recurring decimals: 89 50

113 12

23 32

18 7

5 Calculate the result of this operation. First, transform

the decimals into fractions:

! # d0.18 – 1.89 + 8 n · 1.1 11

6 Zoe spent 1/3 of her money on books and 2/5 on

music. If she has €36 left, how much did she have originally?

Commitment

one-third of the oil in a jug. How many litres of oil were in the jug? How many 3/4-litre bottles can I fill with the rest?

8 One-fifth of the members of a gym are over 60

years old, and two in every three are between 25 and 60 years old.

a) What fraction of the members are 25 years old or younger? b) If there are a total of 525 members, how many are in each age group? 9 I buy a bike and pay in three instalments. In the

first instalment, I pay 3/10 of the total, in the second, I pay 4/5 of the rest, and in the third, I only have to pay €21. How much does the bicycle cost?

10 True or false?

a) All fractions are rational numbers. b) All rational numbers are fractions. c) A fraction is always equivalent to a recurring decimal number. d) A recurring decimal is a rational number.

Watch the video for target 4.a. Think of something you can do to contribute to achieve that goal. Make a commitment to put your idea into practice.

Building Blocks is an educational project of Anaya Educación for Secondary Education with the participation of: K. Chambers, Eleanor Finkenaur, Danny Latimer, R. Oakes, J. Roe, Hannah Peat, Karen Piper, Denise Suárez, Begoña Fuente Larrazabal, Sara Gascón Martín, C. Ordóñez, José Colera Jiménez, Ignacio Gaztelu Albero, Ma José Oliveira González and Ramón Colera Cañas. The following people have worked on this book:

Editorial team: Sara Gascón Martín and Carlos Vallejo INCLUSION AND ANTI-DISCRIMINATION ADVISOR: Víctor Díez Design, technical drawings and maps: Miguel Ángel Castillejos, Miguel Ángel Díaz-Rullo, Patricia G. Serrano, Juan Carlos Quignon

Illustrations: Tatio Viana Layout: DiScript and Isabel Pérez Corrections: Federica Cocco Graphic edition: Olga Sayans Translation: Montero Language Services Photographs: Age Fotostock (AndreaA, Clickalps SRLs, Couperfield, De Agostini / G. Dagl, DEA PICTURE LIBRARY, ePhotocorp, INTERFOTO / Raimund Franken, MOIRENC Camille, Science Source, The Granger Collection, Zoonar/Marek Uliasz), Archivo Anaya (Balaguer, T., Canto, M., Cosano, P., Leiva, Á., Martin, J., Martín, J.A., Osuna, J., Peñuela-Py, E., Pérez de Tudela, M., Pérez-Uz, B., Ramón Ortega, P.-Fototeca de España, Ruiz, J.B., Steel, M., Sánchez, J.), Centro de Investigaciones Sociológicas, Dreamstime (Georgios Kollidas), Getty Images (FatCamera), Instituto Nacional de Estadística, iStock/Getty Images (Albert_ Karimov, Boonchuay1970, Denisfilm, Diane Macdonald, dmitryelagin, gorodenkoff, Halfpoint, JanaShea, JMFPhotograhy, kursatunsal, LuminaStock, MaxRiesgo, milindri, pabradyphoto, Paul-Daniel Florea, Rawf8, scanrail, Sean Pavone, tadamichi, zydesign), NASA, 123RF(Anusorn Phuengprasert Na Chol, donatas1205, Ganna Garmatiy, Ilya Bolotov, Jos Alfonso De Tomas Gargantilla, Kanlayavadee Thephasdin Na Ayuthaya, keltmd, Leonello Calvetti, Marek Uliasz, ostill, Rainer Lesniewski, scanrail, senoldo, Siarhei Nosyreu, stokkete, Vesna Cvorovic).

Academic and Professional Orientation: created in conjunction with Fundación Bertelsmann. Coordinator: Juan José Juárez Calvo. Expert collaborators: Sara Lozano Santiago, Belén Pérez Castro and Pilar Vázquez Hernández.

Commitment to Sustainable Development Goals Our publications contain carefully selected content, illustrations and language to comply with non-discrimination on the grounds of gender, culture or opinion. Grupo Anaya considers social and environmental responsibility to be one of its fundamental values. For this reason, we are committed to: · continually improving our contents and materials related to the environment. · reducing our carbon emissions. · using natural resources responsibly. · making sure that our activity has no negative consequences for endangered forests. These commitments, among others, mean that 100% of the paper used in our books has the PEFC label.

Important information: The activities proposed in this book should be completed in a separate notebook or on sheets of paper, not in the book itself. The links to webpages which appear in this book have been checked before printing. The publisher cannot be liable for any changes or modifications which occur after the date of publication.

© GRUPO ANAYA, S.A., 2021 - C/ Juan Ignacio Luca de Tena, 15 - 28027 Madrid. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publishers.

e d

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DEMO

MATE MÁTIC AS ORIENTADAS A LAS ENSEÑANZAS ACADÉMICAS

3 ESO

Building

Blocks

e rca c e d

MAtemáticas ESO Índice

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D iDA

fraciones y decimales

1 FRACCIONES Una fracción es el cociente indicado de dos números enteros. Dicho cociente puede ser entero 7 – 5 + 2 = 42 o fraccionario a · c = a · c b d b·d 10 12 60

A la unión de todos los números enteros y de todos los números fraccionarios se la llama conjunto de números racionales y se designa por à. Los números racionales son los que se pueden poner en forma de fracción.

Simplificación de fracciones

Recuerda Si el numerador es un múltiplo del denominador, la fracción es un entero. Si no, es una fracción.

Simplificación

Si el numerador y el denominador de una fracción se pueden dividir por un mismo número (distinto de 1 y de –1), al hacerlo diremos que hemos simplificado o reducido la fracción. Observa el esquema ❶.

Fracciones equivalentes Se dice que dos fracciones son equivalentes cuando, al simplificarse, dan lugar a la misma fracción irreducible. 18 y 21 son equivalentes, pues 18 = 18 : 6 = 3 y 21 = 21 : 7 = 3 30 35 30 30 : 6 5 35 35 : 7 5

Comparación de fracciones Dos fracciones con el mismo denominador son muy fáciles de comparar observando sus numeradores. Para comparar dos fracciones con distinto denominador, las «reducimos a común denominador», es decir, buscamos dos fracciones

8 = 4 = –2 –12 –6 3 3 000 = 30 = 2 4 500 45 3

Observa Un procedimiento para comprobar si dos fracciones son equivalentes es el que denominamos productos cruzados: a = c si a · d = c · b b d Por ejemplo, 18 y 21 son 30 35 equivalentes porque: 18 · 35 = 630 = 21 · 30

Practica... 1 ¿A qué fracciones corresponden estos puntos de la

recta? A –3

B –2

–1

C 0

D 1

a) 11> – 4 p orque el primero es positivo y el se5 4 gundo es negativo.

E 2

2 Indica si estas fracciones son o no equivalentes sim-

plificando y mediante productos cruzados: 36 y 78 a) 12 y 21 b) 20 35 102 221

3 ¿Verdadero o falso?

b) 4 > 4 p orque el primero es mayor que 1 y el 5 5 segundo es menor que 1. c) 8 > 7 porque el primero es mayor que 2 y el 3 4 segundo es menor que 2.

Unidad 1

2 OPERACIONES CON FRACCIONES

Suma y resta de fracciones Para sumar (o restar) fracciones con el mismo denominador, se suman (o se restan) sus numeradores y se mantiene el denominador. Para sumar (o restar) fracciones con distinto denominador, se empieza por transformarlas en otras equivalentes con el mismo denominador. Fíjate en el ejemplo ❶.

Producto y cociente de fracciones El producto de dos fracciones es otra fracción cuyo numerador es el producto de sus numeradores y cuyo denominador es el producto de sus denominadores. a · c = a ·c b d b·d

1 7 – 5 +2= 10 12 = 42 – 25 + 120 = 60 60 60 = 42 – 25 + 120 = 137 60 60 2 8 · 7 = 8 · 7 = 56 = 28 3 10 3 · 10 30 15

Fíjate en el ejemplo ❷. El cociente de dos fracciones es el producto de la primera por la inversa de la segunda. a: c =a·d b d b c

3 9 : 5 = 9 · 7 = 63 4 7 4 5 20 4

Fíjate en el ejemplo ❸.

Operaciones combinadas de fracciones Para realizar operaciones combinadas, primero se resuelven los paréntesis y los corchetes, y después, el resto de operaciones, teniendo en cuenta que los productos y los cocientes son anteriores a las sumas y a las restas. Fíjate en el ejmplo ❹.

Fracción de una cantidad Para calcular la fracción, a , de una cantidad, C, multiplicamos a · C. b b Para calcular 3 de 1 200 €, divide entre 5 y luego multiplica el resultado por 3. 5 Fíjate en el ejemplo ❺.

2 – 1 c1 – 1 m + 2 (–2) = 3 3 4 = 2 – 1 · 3 + 2 (–2) = 3 4 3 =2– 1 – 4 = 4 3 24 = – 3 – 16 = 5 12 12 12 12 5 3 · 1 200 € = 5 = 1 200 € · 3 = 720 € 5

Practica.. 1 Calcula y simplifica los resultados.

a) d 3 + 7 – 7 n : 25 b) d 13 – 7 n · d 9 + –13 n 4 6 8 12 15 25 22 33 1 – d 3 – 1n (–3) · d 3 – 1 n 2 4 c) d) 5 3 3 +1 (–2) · d 4 – 6 n 4 3 5

2 Halla la parte del total en cada caso.

a) 1 de 520 000 € b) 7 de 500 edificios 2 10 3 Un ciclista ha recorrido los 5/9 de la etapa de hoy, de 216 km. ¿Cuántos kilómetros lleva recorridos? Contesta y comenta con tus compañeros y compañeras.

3 NÚMEROS DECIMALES La expresión decimal de los números permite valorarlos, compararlos y operar con ellos de forma muy cómoda y eficaz.

Tipos de números decimales • Decimales exactos: tienen un número limitado de cifras.

5,4; 0,97; 8; –0,0725 • Decimales periódicos puros: todas las cifras después de la coma se repiten

periódicamente.

Recuerda

$ 7,81818181... = 7,81 • Decimales periódicos mixtos: tienen otras cifras decimales antes de las cifras

que se repiten.

! 18,35222222... = 18, 352

El grupo de cifras decimales que se repite una y otra vez se llama periodo. Se indica poniendo un arco sobre las cifras correspondientes: ! # 5, 68 16, 147

• Decimales no exactos ni periódicos: son números decimales que tienen in-

finitas cifras que no se repiten periódicamente. Al contrario que los decimales exactos y periódicos, estos números no son racionales, por lo que se denominan números irracionales.

Usa la calculadora En las calculadoras, en lugar de una coma, aparece un punto.

2 = 1,4142135…; π = 3,14159265…

1 437,54 → {∫∫‘¢«|…∞¢}

Practica... 1 Indica qué tipo de número decimal es cada uno de

los siguientes: ! # 2,8 3,52 1, 54 2,7 3,5222…

2 Ordena de menor a mayor.

3 = 1,7320508… π – 2 = 1,1415926…

! ! # a) 3,56; 3,56 ; 3,5 ; 3,56 ! ! # b) –1,32; –1, 32 ; –1,32 ; –1,3 ! c) 2,3; 8 ; 2, 34; 32 ; 21 3 15 10

3 Ordena de mayor a menor los siguientes números:

! 2,35

! 2,5

2,5 2,505005…

4 Escribe tres números comprendidos entre:

! a) 2,5 y 2,5

b) 0,345 y 0,346 ! c) 2,3 y 2,4 d) –4,5 y –4,4

Unidad 1

Transformación de fracciones en decimales Para obtener la expresión decimal de una fracción, se efectúa la división del numerador entre el denominador. El cociente puede ser: • Un número entero: el numerador es múltiplo del denominador.

72 = 8; –240 = –16 15 9 • Un decimal exacto: el denominador de la fracción simplificada solo tiene los

factores primos 2 y 5 (o alguno de ellos).

3,0

20 0,428571 60

se repite

123 = 3,075; 42 = 1,68 25 40

40 50

• Un decimal periódico: el denominador de la fracción simplificada tiene algún

factor primo distinto de 2 y 5.

10 3

A partir de aquí se repiten los cocientes y los restos.

# 86 = 7,# 81 ; 87 = 29 = 1,318 11 66 22

Practica... 5 Comentad en grupo. ¿Verdadero o falso?

! a) 1 = 0,333… = 0,3 3 ! 3 = 3 · 0,333… = 0,999… = 0,9 3 ! Como 3 = 1, resulta que 0,9 = 1 3 ! # b) 5,4 = 5, 44 # # c) 3, 72 = 3,7272727… = 3,727 ! ! d) 0,3 + 0,6 = 1

6 Sin efectuar la división, y atendiendo solo al deno-

minador de la fracción simplificada, di si las siguientes racciones darán lugar a decimales exactos o decimales periódicos: a) 44 150 b) 42 150

c) 101 1024 d) 1001 500

7 Escribe un valor de k para que 84 sea:

a) Un número entero.

Transformación de decimales en fracciones ➜ de decimal exacto a fracción

Expresar en forma de fracción un número decimal exacto es muy fácil, pues el denominador es una potencia de base 10. ➜ de decimal periódico puro a fracción

• Periodo de una sola cifra: N = 5,4 = 5,4444…

10N = 54, 444… 4 10N = 5, 444…

• 2,5 = 25 = 5

10 2 • 3,41 = 341 100 • 0,004 = 4 = 1 1000 250

Observa

Al restar, desaparece la parte decimal.

• Multiplicamos N por una potencia de base 10 para hallar otro número con la misma parte decimal.

10N – N = 54 – 5 → 9N = 49 → N = 49 9 & • Periodo con varias cifras: N = 6,207 = 6,207207207…

• Al restar ambos números, obtenemos un entero.

1 000N = 6 207, 207207… 3 1000N = 6, 207207…

• Despejando N llegamos a la fracción buscada.

Al restar, desaparece la parte decimal: 1 000N – N = 6 207 – 6 → 999N = 6 201 → N = 6 201 999 ➜ de decimal periódico mixto a fracción

• Para transformar N = 2,563 en una fracción:

Observa

N = 2,5636363… Multiplicamos por 10 para obtener un decimal periódico puro. 10N = 25,636363… Ahora, multiplicamos por 100 para obtener otro con la misma parte decimal. 1 000N = 2 563,636363… Al restar este al anterior, desaparece la parte decimal. Es decir, se obtiene un número entero. 1 000N – 10N = 2 563 – 25 → 990N = 2 538 → N = 2 538 990

• Multiplicamos N dos veces por potencias de base 10 para obtener dos decimales periódicos puros con el mismo periodo. • Al restarlos se obtiene un número entero. • Despejando N se obtiene la fracción buscada.

Practica... b) Un decimal exacto. c) Un decimal periódico. 8 Expresa en forma de fracción.

a) 6,2 b) 0,63 c) 1,0004 ! ! ! 0,1 f ) 2,7 d) 3,5 e) & & # g) 0,23 h) 41,041 i) 40,028 ! & # j) 5,9 k) 7,009 l) 0,99

9 Expresa como fracciones los siguientes decimales:

! ! # 0, 001 c) a) 6, 25 b) 5, 018

Unidad 1

4 FRACCIONES Y DECIMALES CON LA CALCULADORA

Fracciones

Usa la calculadora Para introducir las fracciones en la calculadora utilizamos la tecla flechas ”’‘“. Escribir 3 → 4

3 3 ’ 4 ” 4

6 → 8

3 Si pulsamos = 4

6 6 ’ 8 ” 8

y las

3 4

Observa

6 = 8

3 4

Puedes pulsar = para simplificar una fracción.

Operaciones con fracciones

Practica...

Usa la calculadora

1 Introduce en la

Por ejemplo, obtengamos 2 + 5 · 1 – 11 : 3 6 12 2 ’ 3 ”+ 5 *

1 ’ 6 ”-

2 1 11 + 5 x – 11 ’ 12 ”= 3 6 12

7 12

Decimales

Usa la calculadora Los decimales no periódicos se escriben de forma natural teniendo en cuenta que en lugar de la coma, se debe poner un punto (.).

La tecla �, aplicada a un número obtenido en la SALIDA, lo transforma de fracción a decimal, o viceversa. 3,875 → 3 . 875 =

3,875 31 8

�

3,875

# 5, 491 → 5 . 4

91 =

5,491 5437 990

�

c) 27 15 e) 0,27

d) 3,25 f ) 0,321

2 Haz con la calculadora

estas operaciones. Obtén los resultados en forma de fracción y de número decimal: a) 5 – 2 4 7

3,875

Para escribir un decimal periódico, usaremos las teclas

calculadora estas expresiones y comprueba que al pulsar =, se simplifican las fracciones o se obtienen las fracciones correspondientes: 8 a) 3 b) 12 5

b) d–3 + 1 n : 2 3 5

c) d –2 – 3 n – 2 5 7

5,491 5,491

d) 2 – d 1 + 3n : 1 7 8 3

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Equipo de edición: Virginia Álvarez, M.a Federica Cocco, Beatriz Robles y Elia Ureña (adaptación del libro Matemáticas 3.° ESO)

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