Issuu

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 Albero u l e t z Ga nez, I. é m i J s ra a Caña r e J. Cole l o C R.

Building

Blocks

this is

your book

Reading and listening

CH UNIT

F EA THE OPENING PAGES O

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

number of writing the same Different ways same number: ways of writing the These are three different

NATURAL NUMBERS g

Reading and listenin

is 3 Which number

throughout histor

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

that follows in all three it? Write the number system? the decimal numeral this number? other ways to write

Can you think of any

ds lication metho It shows multiplication below. of an ancient Indian Look at the example 346 × 57. numbers how they calculated with the individual • They used a table 5 6 along the sides. 0 7 was multiplied top the on 4 3 was • First, each number 0 4 2 the side. The answer , 3 2 by each number along nding box. For example 1 5 2 8 2 written in the correspo shows: 1 2 the yellow box 12 1 4 × 7 = 28 9 6 were in each vertical column 1 2 • Then, the numbers could only be one digit in added together. There 1 9 7 2 2 each column. cations: for the following multipli 6 Use this method b) 453 × 26 a) 208 × 34

Different multip

ions Combined operat pay a €20 centre, you have to at your local sports 7 To play sports €15 per month. below: registration fee and ns to the descriptions g mathematical expressio Match the followin 15 · 3 + 15 · 3 20 3 · (20 + 15) pay in the second term. a) The amount you in the first term. pay you amount b) The for three children. pay in the first month c) The amount you

GE BANK GE BANK LANGUA LANGUA BANK GE NK GE BANK GE BA LANGUA 9 LANGUA LANGUA BANK GE BANK GE NK GE BANK GE BA LANGUA LANGUA LANGUA LANGUA BANK GE BANK NK GE UAGE BA LANGUA LANGUA

LANG

CONTENT DEVELOPMENT AND ACTIVITIES

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.

Unit 1

LARGE NUMBERS

ROUNDING NATURAL NUMBERS

Many numbers contain more than nine figures. For example, there are 7 000 000 000 people on the Earth, 3 153 600 000 seconds in a century and 9 460 800 000 000 kilometres in a light year.

ones

tens

hundreds

thousands

…

An uncommon name for a thousand millions (1 000 000 000) is a milliard. Sometimes, the prefix giga is also used. For example: 1 000 000 000 bytes = 1 gigabyte

millions

billón

thousands of millions

The decimal numeral system allows us to write numbers that contain as many figures as we want. The table below shows the place value for some numbers with more than 9 figures:

Interesting fact

When a number has many figures, it is difficult to remember and makes calculations more difficult. We often replace it with an approximate value ended in zeros. That number is more manageable. For example: There are 31 853 000 €500 notes in circulation in Spain.

F ocus on English In English, we write the € symbol before the amount (not after it like in Spanish). However, when we say the amount, we say Euro after it. For example: €500 ↔ Five hundred Euros.

Approximately, how many thousand million Euros is this?

31 853 000 × 500 = 15 926 500 000 Approximately, they are sixteen thousand millions. Rounding is the most frequent and easy method of approximation. To round a number to a specific place value: • We replace with zeros all the numbers to the right of that place value. • If the first number being replaced is bigger than or equal to five, we add one

6 GREATEST COMMON DIVISOR

F ocus on English

The icons included with some activities indicate the keys to the project.

Consolidating

more unit to the following figure.

ideas

Unit 3

The universe was formed A young person’s brain The volume of the Earth thirteen thousand eight contains around one is approximately one Calculating the greatest hundred million years ago.ãhundred thousand millions billón cubic common kilometres. of neurons. method) (optimal

Help

1 Use the diagram below to help you ideas round the number 384 523 to the idating Consol The English words billions, trillions, nearest hundreds of thousands; to the nearest tens of thousands; and to the divisor quadrillions… are false friends. They nearest thousands. 3 Copy and complete: do not mean the same in Spanish as hundreds of thousands tens of thousands thousands in English. 40 = 2 $ 2 $ 2 $ 5 The method you have learnt on the previous page is useful for simple numbers. (40, 50) 3 83 348 854 425 532 23 3 3 8=3 2348 $ 8545 $4255 532 233 GCD 3 3 83 =348 854 ·425 532 =23… 3 For example: 50 However,byfor larger numbers it is easier to decompose them into prime factor. • One million ↔ A 1 followed 6 zeros. +1 +1 +1 H8Th ≥85≥85≥ 5 = = = T Th4 <345<45< 5 +1 +1 +1 Th 5 ≥55≥55≥ 5 1 billion ↔ A 1 followed by 9 zeros. Here isby an12 example: 54 = 2 $ 3 • One billón ↔ A 1 followed zeros. 4 GCD (54, 90) = 2 · 3 = … 1 trillion ↔ A 1 followed by ... ...0...0 0 0 0 0 0 ... ... ...0 0 0 0 0 90...=...20...$ 3020$ 50 0 0 Example • One trillón ↔ A 1 followed by 18 zeros. Traditional method 12 zeros. Let’s practise!

Calculate the GCD (40, 60).

Divisors of 40

1 2 4 5 8 10 20 40 1 Read the first paragraph Then, write 3 4this 10 12 15 20 30 the 60 1 2 on 5 6page.

following numbers in words:

Divisors of 60 a) The number of people on Earth. GCD (40, 60) = 20 b) The number of seconds in a century.

c) The number of kilometres in a light year. 2 Write the following numbers in figures:

a) Twenty-eight million three hundred and fifty thousands. anayaeducacion.es GeoGebra. b) One hundred and forty-three millions. Calculate the GCD of two numbers.

c) Two thousand seven hundred millions. d) Sixteen gigas. e) One and a half billón.

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

Remember

• Step one: decompose into prime factors. 4 0 2in your notebook: 3 Copy and complete

Let’s practise!

If we round the number 52 722: – to the nearest tens of Example thousands → 50 000 Calculating the greatest common – to the nearest thousandsdivisor → 53 of 00060 and 72:

60 = 2 · 2 · 3 · 5 72 = 2 · 2 · 2 · 3 · 3

60 = 2 2 $ 3 $ 5 72 = 2 3 $ 3 2 90 = 2 · · 4 Round the following numbers to the nearest millions: GCD (60, 72) = 22 · 3 = 12 315356 = 3000· · b) 36 905 000 a) 24 c) 274 825 048 GCD (90, 315) = … 5 The sign below shows the price of a house in Euros. €138 000 FOR SALE €138 300 138 290 € €138 290 €140 000

4 Copy and complete to calculate the greatest common divisor of 90 and 315:

9 0 3 1 5 2 1 Round the following numbers to the nearest thousands:

6 0 2 a) 24 963 b) 7 280 c) 40 274 d) 99 834 2 0 thousands 2 3 0 2 a) One thousand = one ... 1 0 millions 2 1 5 3 60 = 22 · 3 · 5 2 Round these numbers to the nearest hundreds and to 40 = 23...· 5 b) One thousand = one the nearest tens of thousands: 5 5 5 1 1 c) One million5of millions = one ... a) 530 298 b) 828 502 c) 359 481 d) 29 935 236 1 1 4 The human body contains between ten and seventy 3 Look at the newspaper in the picture. Round the • Step two: select the prime factors of the GCD. million of millions cells. Express both amounts in number of tourists toLet’s the practise! nearest millions and the a) Which of the three approximations is closest to the billones. Remember that the greatest common divisor must be a divisor of 40 and 60.amount It they spent to the nearest thousands of millions. real value? must also be the greatest number possible. We must take: 1 Look and complete. 6 Calculate: 5 How do you say the number that is written as a 1 b) Which approximation would you use in an In 2018, 40 = 2 · 2 · 2 · 5 Divisors of 24 → 1 2 3 4 6 informal 8 12 conversation? 24 a) GCD (20, 24) b) GCD (24, 36) followed bycommon 30 zeros?factors of 40 and 60. — The 82 600 000 visited tourists . They 60 = 2 · 2 · 3 · 5 Spain 8 Divisors of 30 → 1 2 3 65 The 6 10 89 67 6 Scientists estimate that there are three cuatrillones spent town15hall30has a budgetc)ofGCD €149(54, 63760) to refurbish d) GCD (56, 70) Euros. n millio kilograms water in number our seasofand oceans. What do GCD (24, 30) = … e) GCD (120, 144) would f ) GCD (140, 180) its sports centre. Which approximate number — Theofmaximum factors you think a cuatrillón is? you use to tell a friend about this? possible. GCD (40, 60) = 2 · 2 · 5 2 Calculate using the method from the previous exercise: 7 Calculate the GCD (a, b). What do you notice? • Step three: calculate the GCD.

When one of the numbers is a multiple of the other, the GCD is the lowest of the two. Example: GCD (15, 30) = 15 Check. 15 = 3 · 5 30 = 2 · 3 · 5 GCD (15, 30) = 3 · 5 = 15

GCD (40, 60) = 2 · 2 · 5 = 20 To obtain the greatest common divisor of several numbers: 1st We decompose them into prime factors. 2nd We only take the common prime factors, each one raised to the smallest exponent possible. 3rd We multiply the chosen factors.

a) GCD (10, 15)

b) GCD (12, 18) a) a = 4numbers.b) a = 5 anayaeducacion.es Practise rounding

c) GCD (16, 24)

d) GCD (30, 45)

b=8

a) GCD (3, 9)

b) GCD (6, 9)

c) GCD (30, 40)

d) GCD (50, 75)

9 0 4 5

2 0 0 1 0 0 5 0 2 5 5 1

2 2 2 5 5

2 6 0 1 3 0 6 5 1 3 1

GCD (200, 260) = 22 · 5 = 20

2 2 5 13

A warehouse packs and distributes 200 kg of apples and 260 kg of oranges. It uses boxes that weigh the same. It distributes the largest load possible. How many kilos does it put in each box? The weight of the boxes is a common divisor of 200 and 260. It is the greatest possible divisor, in other words, the greatest common divisor. 200 : 20 = 10 boxes of apples GCD (200, 260) = 20 kg * 260 : 20 = 13 boxes of oranges Answer: Each box weighs 20 kg. It packs 10 boxes of apples and 13 boxes of oranges.

d) a = 6 b = 18

b) a = 23 · 52 b = 22 · 52 · 7

and a 60-litre barrel of sunflower oil. She puts the oil in jars of the same size. They are as large as possible. She does not mix the oils. How large are the jars?

10 A carpenter has two pieces of wood measuring

5 Calculate the GCD (a, b) of each case:

b = 2 · 5 · 11

b = 12

9 A restaurant owner buys an 80-litre barrel of olive oil

1 0 0 5 0

_ 60 = 2 ·…b GCD (60, 90) = … b 90 = 2 ·…` GCD (60, 100) = … 100 = 2 ·…b GCD (90, 100) = … a a) a = 3 · 5 · 11

c) a = 4 13

You draw the largest grid possible. None of the squares in the grid are divided. What is the size of the squares?

4 Copy in your notebook and calculate:

6 0 3 0

b = 10

Consolidating Ideas. Exercises to co mplete and consolidat e the theory that the teacher has explained to you.

8 You have a piece of paper that measures 30 cm × 21 cm.

3 Solve without a calculator:

Problem solved

Calculating the GCD (200, 260)

ways.

use 4 Do any of them

s times. All s since prehistoric to express quantitie Romans, We use numeral systems the Egyptians, Greeks, Each ons used them, from , China and India. the major civilisati to the people of Babylon These systems were then Arabs and Mayans its own numeral system. time. d over civilisation develope to village and changed sheep, fruit like passed on from village products natural only used to count , people simple. For example At first, numbers were very were systems and coins. Early numeral cut marks on sticks. or drew hands and fingers, : they used letters instead r way of writing numbers different value. Imagine Romans had a particula had a use today! Each letter ons. For example: of the numbers we use letters in calculati how the Romans would XXIV MCCCXLVI + DCCCX difficult! would be even more ted? Multiplications numeral Does it seem complica the Hindu-Arabic became simpler with numbers Luckily for us, things It introduced decimal positional notation. system, which we call of the system we use today. bases counting more and established the , but they made specific er were more complex rememb , systems new The write numbers to systems help us to e, we must be able practical. Numeral them to others. Therefor in their written form. amounts and express see them calculations when we Try understand and solve of numeral systems. you can find the concept text the of title 1 In the own words. quick to define it using your a is here r, numerals. Howeve familiar with Roman 2 You are already of each letter: reminder of the value

Numeral system

1 000 500 100 50 10 5 1 numeral the text into the decimal numeral Roman addition in into the Roman Now, translate the n and translate it back operatio the do system, system.

divided Each unit is phs and into epigra hs. subepigrap portant The most im in bold. re a contents

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

c) a = 22 · 7 · 13 b = 2 · 32 · 13

180 cm and 240 cm long. He cuts both of them into equal sections. The sections are as long as possible, without wasting wood. How long is each section?

Let’s think! 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 “Portfolio”, resource bank portfolio. duacion.es create your In the anayae on how to guidance you will find

Unit 1

PROBLEM EXERCISES AND

Numeral systems

Exercises and problems. For you to apply all the contents that you learnt throughout the unit.

Translate the following Egyptian numbers into the decimal numeral system: a

Write the following numbers using the Egyptian additive system: a) 48

Glossary

b) 235

c) 2 130

Write the following in Roman numerals: a) 87

b) 425

c) 2 600

How many figures are there in a billón? And in a trillón? How many zeros are there in each number?

The table below contains data on vegetable consumption in Spain in 2016: (thousands of €)

6 195 054

fresh fruit

4 369 449

vegetables and potatoes

3 626 510

5 214 031

total

7 995 959

11 409 085

b) One hundred millions is equal to one thousand

What can numbers tell us?

1. NATURAL NUMBERShundreds of thousands.

factorise divisor (GCD)

not hundred depend gigas on how sumands are grouped. d) One are the equal to one billón. billón

a) Which one is at the end of the corridor? b) Which one is on the top floor? c) Which ones are on the same floor?

A 1 followed by 12tozeros (One million millions). e) One billón is equal one million of millions.

lowest common

8 2

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

5 7

Mental arithmetic. a) 5 + 7 – 3 – 4 c) 10 – 6 + 3 – 7 e) 12 + 13 + 8 – 23

5 6

12 Here are some hotel room numbers: 401;greatest 235; common c) One thousandoftimes a million equal to one giga. A property addition and is multiplication that says that the 724;result 231. of the sum does

associative property

8 b) 651 + 283 – 459 d) 1 648 – 725 – 263

Copy, calculate and complete in your notebook: a) 48 + … = 163 b) … + 256 = 359 c) 628 – … = 199 d) … – 284 = 196

Copy the table in your notebook, but rounding the figures to the nearest million tonnes and to the nearest hundred million Euros.

True or false? a) One million is equal to one thousand hundreds.

Calculate: a) 6 070 + 893 + 527 c) 831 – 392 – 76

value

(tonnes)

Copy and complete in your notebook:

Addition and substraction

You see an advert for a house that costs €293 528. You tell a friend about it a few days later, but you can’t remember the exact price. Which of the following sentences do you use instead? Why? a) It costs almost three hundred thousand Euros. b) It costs just over two hundred thousand Euros. c) It costs two hundred and ninety thousand Euros.

weight

Write the number fifty-seven using at least three different numeral systems.

Target 11.c. According to a Cairo newspaper, the population of Egypt’s capital city was 19 487 245 in June 2018. If somebody asked you for the approximate population of Cairo, what would you say? If the population of Cairo keeps growing, what will be its population by 2030? What measures would you take for Cairo to become a sustainable city by 2030?

d) 54 528

Operations

6 22

b) 18 – 4 – 5 – 6 d) 8 + 5 – 4 – 3 – 5 f ) 40 – 18 – 12 – 6

Copy and complete in your notebook: a) 123 · … = 5 904

b) … · 86 = 1 548

c) … : 57 = 26

d) 1 862 : … = 133

Calculate: a) 47 – (35 – 28) b) 52 – (36 – 27) c) 128 – (86 – 45 – 12) d) 237 – (152 + 48 – 14) e) 348 – (148 – 86 + 29) f ) 235 – (340 – 152 – 84)

Mental arithmetic.

a) 3 · (10 : 5)

b) (4 · 6) : 8

c) 20 : (2 · 5)

d) (30 : 5) · 3

e) 10 : (40 : 8)

f ) (40 : 8) : 5

Calculate the following in your head. Remember that dividing by 5 is the same as dividing by 10, and then multiplying by 2. :5

• 90 : 10

18 ·2

a) 60 : 5 b) 80 : 5 c) 120 : 5 Calculate: d) 140 : 5 e) 170 : 5 f ) 200 : 5 a) 5 – [7 – (2 + 3)] b) 3 + [8 – (4 + 3)] g) 210 : 5 h) 340 : 5 i) 420 : 5 c) To 2 +express [6 + (13a–number 7)] as a product of its divisors. 25 Copy and complete in your notebook: d) The 7 – [12 – (2 +of5)]the common divisors of two or more numbers. greatest 6 · (8 + 2) = 6 · 8 + 6 · 2 = 60 e) 20 – [15 – (11 – 9)] … = 5 · 9 – 5 · 6 = .... f ) 15 – [17 – (8 + 4)] The lowest of the common multiples of two or more numbers.

(10 – 8) · 4 = … = .... multiple (LCM) Check your answers: To join several quantities (addends) into one. … = 7 · 12 – 2 · 12 = .... multiple an exact number of times. a) Number 3; b) 4; containing c) 14; d) 2;another e) 7; f )number 10 Star A is five light years away. Star B is five billones Numeral system where adding symbols adds their represented amount. prime Number that is only divisible by itself and the unit. Which property did you use? kilometres away. Which star is the furthest? 13 Do you remember how car number plates are number Multiplication and division A property of addtion and multiplication that says that the result of the sum does ordered? Have a look at these plates: 26 Mental arithmetic. not change if the order of the sumands changes. Rounding Multiply: 19 3894 a) One barrel can hold 5 litres of water. How many decimal numeral system Positional numeral system with ten symbols or figures (0, E1, 2,3948 3, 4,FBG 5, 6, 7, 8Eand 9). FBG E 4389 GFB a) 16 · 10 b) 128 · 10 c) 60 · 10 8 Copy and complete the table in your notebook: barrels can you fill with 100 litres of water? This is the numeral system we currently use. a) Which plate is the oldest? And the newest? 4. INTEGERS d) 17 · 100 e) 85 · 100 f ) 120 · 100 distributive property A property of multiplication that says that the product of the multiplication does not b) One kilo of almonds costs €12. How much do rounding b) Which is the plate number directly after the red g) The 22 · natural 1 000 number h) 134 000from i) 140 · 1 000 absolute value we· 1get removing its sign. change if we remove the brackets. you pay for 5 kilos? to the nearest to the nearest one? And the previous one? number of an integer hundreds thousands millionequal parts. 20 Calculate the quotient and remainder: division The distribution of of a whole among several, c) There are 24 cans of soft drink in a box. How c) How many plates were made between the red and opposite of an integer same absolute opposite sign. 2 830 554 a) Another 2 647 : 8 integer with the b) 1 345 : 29 value, but with the many cans are there in 10 boxes? exact division Division where the remainder is zero. the green ones? 19 270 000 Z set zero, and the negatives of the natural c) The 9 045set : 45of all positive natural d) 7 482numbers, : 174 integer division Division where the remainder is not zero. d) How many cars had the same letters as the blue d) It costs €360 to replace all four tyres on a car. numbers. 399 675 000 plate after it? e) 7 971 : 2 657 f ) 27 178 : 254 How much does each tyre cost? million A 1 followed by 6 zeros. addition

additive system

commutative property

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

multiplication natural numbers

A repeated addition of the same value. 20

Numbers that can be used to count items.

numeral system

Set of symbols and rules used to represent numbers.

5. DECIMALS

positional numeral system

Numeral system where symbols have different values depending on their level.

subtraction

decimal number

To remove an amount (subtrahend) from another (minuend) to find out the difference between the two.

The result of a non-exact quotient. It has an integer portion and a decimal portion, separated by a decimal point.

hundredth

The result of dividing one tenth into ten equal parts.

number line

A one-dimensional line that contains all the real numbers.

tenth

The result of dividing one into ten equal parts.

thousandth

The result of dividing one hundredth into ten equal parts.

unit

The element used to build all natural numbers, represented by the number 1.

trillón

A 1 followed by 18 zeros (One million billones).

2. POWERS AND ROOTS power

A shortened form of writing a product of equal factors.

power of base 10

The unit followed by as many zeros as figures marked in the exponent.

product of powers with the same base

To multiply two powers with the same base, we keep the same base and add the exponents together.

quotient of powers with the same base

To divide two powers with the same base, we keep the same base and subtract the exponents.

3. DIVISIBILITY

6. THE METRIC DECIMAL SYSTEM angstrom

A unit used to measure atomic distances.

astronomical unit

The average distance from the Earth to the Sun. It is used to measure the distance between planets.

gram

The main unit for measuring masses.

light year

The distance light travels in one year. It is used to measure the distance between galaxies.

divisible

Number that when divided by another gives an exact result.

litre

The main unit for measuring capacities.

composite number

A number that can be factorised into simpler factors.

magnitude

Quality and property of objects that can be measured and quantified numerically.

divisor

Number that is contained in another number an exact number of times.

metric decimal system

The set of units of measurement for basic magnitudes.

factor

Each of the quantities that can be multiplied to form a product.

micrometre

One thousandth of a millimetre.

290

291

MATHS WORKSHOP Unit 1

MATHS WORKSHO READ AND LEARN

PRACTICE MAKES PERFECT! The history of counting

Try it out

Would we be able to live in the modern-day world without numbers? The birth and development of civilizations was always hand in hand with the birth and development of numbers. Over time, complex tools were developed to represent and make calculations with numbers.

• Copy the diagram below in your notebook. Write the numbers 1 to 9 in each box. You can only use each number once. Each set of three numbers along each line should add up to 15.

Hands were used as early calculators. Later, people developed methods for working with larger numbers. For example, they used piles of stones, beads on strings, abacuses… People invented mechanical calculators in more recent times. Electronic calculators and computers are even more modern. They are capable of managing enormous numbers and can do calculations almost instantly! Multiplication in ancient Egypt Ancient Egyptians multiplied numbers by two, then by two again and again, until they reached the number they wanted. For example, this is how they would calculate 23 × 18. They made two columns of numbers following these rules: — In the first column, they doubled the number 1 until they reached the first number in the calculation without going over it. In this case, without going over 23. ←• 1 ⎯→ ←• 2 ⎯→

— In the second column, they doubled the second number, 18 in our example, the same number of times as they doubled the number 1 in the first column.

18 →

— Then, they added all the numbers they needed from the first column together to make 23: 1 + 2 + 4 + 16 = 23

36 → ←• 4 ⎯→ 72 → 8 144

— Finally, in the second column, they added the same rows of numbers together as they did in column one. This gave them the solution of the multiplication: 18 + 36 + 72 + 288 = 414 → 23 × 18 = 414

←• 16 ⎯→ 288 → 414 ← → 23

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

• Use this method to solve the following multiplications: a) 17 × 41

b) 41 × 17

INVESTIGATE

• Work out how they moved the beads to do this calculation.

a) 211 + 42

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

Not possible

2 7

15 16 17 18 19

528

Which systems are additive? Which systems are positional? What is the difference? 2 Copy and fill in the blanks in your notebook:

a) 18 ·

= 180

c) 4 000 :

= 40

a) 154 ·

= 462

c) 30 275 :

· 100 = 27 000

: 10 = 38

= 35

· 125 + 8

a) 12 + 3 · 5 – 2

b) 19 – 5 · (10 – 7) + 4 · 7

c) 7 · 3 – 4 · 2 + 2

d) 10 · [7 · 5 – (4 + 6 · 3)]

more chairs than stools. How many chairs are there? How many stools?

Commitment

a) Write the first number in figures and the second number in words. b) Round them to the nearest tens of thousands. c) Round them to the place value you think is most appropriate for the information given. Explain your decision. 7 A van travelling at 60 km/h passes a car travelling at

90 km/h in the opposite direction.

What is the distance between them after ten minutes?

: 27 = 98

d) 1 508 =

6 Read the following statements:

• In April 2018, the world population was 7 601 767 200.

10 11 12 13 14

mayan decimal

anayaeducacion.es Answer key.

0 1

b) 131 – 6

triangles

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

• Brazil has a surface area of eight million five hundred and fourteen thousand eight hundred and seventy-seven square kilometres.

numeral systems

egyptian

remaining corners

SELF-ASSESSMENT 1 Complete the following table in your notebook:

5 There are 60 seats in a cafe. There are three times

• Then, draw diagrams to show how you would use the abacus for the following calculations:

squares

4 Solve the following combined operations: +15

Complete the table below to help you:

corners

• How many three figure numbers can you make using only the numbers 1, 2 and 3?

3 Copy and complete in your notebook:

Many cultures used abacuses throughout history. One of the most effective was the Chinese abacus. The following images show how people used it to calculate 326 + 15:

• There are several square sandwiches on a plate. We cut some of them in half, to make them triangular. After doing this, there are 18 corners in total. How many sandwiches do we cut in half and how many are whole?

8 A beekeeper has 187 hives. She collects two harvests

each year. Each hive produces approximately 9 kilos of honey in every harvest. a) The honey is placed into half kilo jars. How many jars of honey does she produce each year? b) The jars are put into boxes. There are six jars in each box. Each box is sold for €18. What is the beekeeper’s annual profit? c) Round this number.

Watch the video for target 13.3. Think of something you can do to contribute to achieve that goal. Make a commitment to put your idea into practice.

ment. Self-assess ies, ese activit th By doing r u o y heck you can c how ding and n ta unders t. rn a le e v ha much you

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, etc.). 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 Key concepts Activities with GeoGebra Learn by playing Problem solving practice Glossary 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 NATURAL NUMBERS

5 Page 8

1. Numeral systems.................................................................. 2. Large numbers...................................................................... 3. Rounding natural numbers............................................... 4. Basic operations with natural numbers....................... 5. Expressions with combined operations...................... Exercises and problems......................................................... Maths workshop........................................................................ Self-assessment........................................................................

POWERS AND ROOTS

DIVISIBILITY

INTEGERS

13 14 18 20 26 27

30 32 33 36 39 42 43

Page 44

1. The relation of divisibility................................................. 2. Multiples and divisors of a number............................... 3. Prime and composite numbers...................................... 4. Decomposing a number into its prime factors...................................................................................... 5. Lowest common multiple................................................. 6. Greatest common divisor................................................. Exercises and problems......................................................... Maths workshop........................................................................ Self-assessment........................................................................

Page 28

1. Powers...................................................................................... 2. Powers of base 10: Uses.................................................... 3. Operating with powers...................................................... 4. Square roots.......................................................................... Exercises and problems......................................................... Maths workshop........................................................................ Self-assessment........................................................................

46 48 51 52 54 57 60 62 63

Page 64

1. Positive and negative numbers...................................... 2. The set of integers............................................................... 3. Addition and subtraction with integers...................... 4. Addition and subtraction with brackets..................... 5. Multiplication and division with integers.................... 6. Combined operations......................................................... 7. Powers and roots of integers.......................................... Exercises and problems......................................................... Maths workshop........................................................................ Self-assessment........................................................................

66 68 70 72 75 77 78 80 84 85

DECIMALS

Page 86

1. The structure of decimal numbers................................ 88 2. Addition, subtraction and multiplication with decimals......................................................................... 92 3. Dividing decimals................................................................. 94 4. Square roots and decimal numbers.............................. 97 Exercises and problems......................................................... 98 Maths workshop........................................................................ 102 Self-assessment........................................................................ 103

THE METRIC DECIMAL SYSTEM

Page 104

1. Magnitudes and measurements..................................... 106 2. The metric decimal system.............................................. 107 3. Units of measurement for fundamental magnitudes............................................................................ 108 4. Conversion of units............................................................. 110 5. Complex and simple amounts........................................ 111 6. Measuring surface areas.................................................... 112 Exercises and problems......................................................... 116 Maths workshop........................................................................ 120 Self-assessment........................................................................ 121

FRACTIONS

Page 122

1. What are fractions?............................................................. 124 2. The relationship between fractions and decimals.. 126 3. Equivalent fractions............................................................ 127 4. Problems with fractions.................................................... 130 Exercises and problems......................................................... 131 Maths workshop........................................................................ 134 Self-assessment........................................................................ 135

OPERATING WITH FRACTIONS

Page 136

1. Reducing to a common denominator.......................... 138 2. Adding and subtracting fractions.................................. 140 3. Multiplying and dividing fractions................................. 142 4. Combined operations......................................................... 144 5. Problems with fractions.................................................... 145 Exercises and problems......................................................... 146 Maths workshop......................................................................... 150 Self-assessment........................................................................ 151

PROPORTIONALITY AND PERCENTAGES

Page 152

1. Proportionality between magnitudes.......................... 154 2. Direct proportionality problems.................................... 156 3. Inverse proportionality problems.................................. 158 4. Percentages........................................................................... 160 5. Percentage increases and decreases........................... 163 Exercises and problems......................................................... 164 Maths workshop........................................................................ 168 Self-assessment........................................................................ 169

ALGEBRA

Page 170

1. Letters instead of numbers.............................................. 172 2. Algebraic expressions........................................................ 175 3. Equations................................................................................ 179 4. First methods for solving equations............................. 180 5. Solving first-degree equations with one unknown.................................................................................. 182 6. Solving problems through equations........................... 186 Exercises and problems......................................................... 188 Maths workshop........................................................................ 192 Self-assessment........................................................................ 193

LINES AND ANGLES

Page 194

1. Basic elements of geometry............................................ 196 2. Two important lines............................................................ 198 3. Angles...................................................................................... 199 4. Angle measures.................................................................... 200 5. Operating with angle measures..................................... 202 6. Angular relationships......................................................... 204 7. Angles in polygons.............................................................. 205 8. Angles in a circumference................................................ 206 Exercises and problems.......................................................... 208 Maths workshop......................................................................... 210 Self-assessment......................................................................... 211

GEOMETRIC SHAPES

Page 212

1. Polygons and other plane shapes................................ 214 2. Symmetries in plane shapes........................................... 215 3. Triangles................................................................................. 216 4. Quadrilaterals....................................................................... 218 5. Regular polygons and circles......................................... 220 6. Cordovan triangle and related figures........................... 222 7. Pythagorean theorem....................................................... 224 8. Applications of the Pythagorean theorem............... 225 9. Geometric shapes............................................................... 228 10. Polyhedra............................................................................... 229 11. Solids of revolution............................................................. 230 Exercises and problems......................................................... 231 Maths workshop........................................................................ 236 Self-assessment........................................................................ 237

AREAS AND PERIMETERS

Page 238

1. Measuring quadrilaterals................................................... 240 2. Measuring triangles............................................................. 242 3. Measuring polygons............................................................ 243 4. Measuring circles................................................................. 244 5. The Pythagorean theorem for calculating areas..... 246 Exercises and problems......................................................... 248 Maths workshop........................................................................ 254 Self-assessment........................................................................ 255

GRAPHS OF FUNCTIONS

Page 256

1. Cartesian coordinates........................................................ 258 2. Points that provide information..................................... 259 3. Points that are related....................................................... 260 4. Interpreting graphs............................................................. 262 5. Linear functions. Equation and representation........ 265 Exercises and problems......................................................... 266 Maths workshop........................................................................ 270 Self-assessment........................................................................ 271

STATISTICS

Page 272

1. Statistical analysis process............................................... 274 2. Frequency and frequency tables................................... 276 3. Statistical graphs................................................................. 278 4. Statistical parameters........................................................ 280 Exercises and problems......................................................... 284 Maths workshop........................................................................ 288 Self-assessment........................................................................ 289

CHANCE AND PROBABILITY

Page 290

1. Random events..................................................................... 292 2. Probability of an event....................................................... 294 3. Assigning probabilities to regular experiments....... 296 4. Some strategies for calculating probabilities........... 298 Exercises and problems......................................................... 300 Maths workshop........................................................................ 304 Self-assessment........................................................................ 305

Annex • Glossary................................................................................. 306

NATURAL NUMBERS Reading and listening

Numeral systems throughout history We use numeral systems to express quantities since prehistoric times. All the major civilisations used them, from the Egyptians, Greeks, Romans, Arabs and Mayans to the people of Babylon, China and India. Each civilisation developed its own numeral system. These systems were then passed on from village to village and changed over time. At first, numbers were only used to count natural products like sheep, fruit and coins. Early numeral systems were very simple. For example, people drew hands and fingers, or cut marks on sticks. Romans had a particular way of writing numbers: they used letters instead of the numbers we use today! Each letter had a different value. Imagine how the Romans would use letters in calculations. For example: MCCCXLVI + DCCCXXXIV Does it seem complicated? Multiplications would be even more difficult! Luckily for us, things became simpler with the Hindu-Arabic numeral system, which we call positional notation. It introduced decimal numbers and established the bases of the system we use today. The new systems were more complex, but they made counting more practical. Numeral systems help us to write numbers, remember specific amounts and express them to others. Therefore, we must be able to understand and solve calculations when we see them in their written form. 1 In the title of the text you can find the concept of numeral systems. Try

to define it using your own words.

2 You are already familiar with Roman numerals. However, here is a quick

reminder of the value of each letter:

100

500

1 000

Now, translate the Roman addition in the text into the decimal numeral system, do the operation and translate it back into the Roman numeral system.

Different ways of writing the same number These are three different ways of writing the same number:

3 Which number is it? Write the number that follows in all three ways. 4 Do any of them use the decimal numeral system? 5

Can you think of any other ways to write this number?

Different multiplication methods Look at the example of an ancient Indian multiplication below. It shows how they calculated 346 × 57. 6

3 0 7 3 2 0 4 2 1 5 2 8 2 2 1

9 6 1 2 1 9 7 2 2

• They used a table with the individual numbers along the sides. • First, each number on the top was multiplied by each number along the side. The answer was written in the corresponding box. For example, the yellow box shows: 4 × 7 = 28

• Then, the numbers in each vertical column were added together. There could only be one digit in each column.

6 Use this method for the following multiplications:

a) 208 × 34

b) 453 × 26

Combined operations 7 To play sports at your local sports centre, you have to pay a €20

registration fee and €15 per month.

Match the following mathematical expressions to the descriptions below: (20 + 15) · 3

20 + 15 · 3

15 · 3

a) The amount you pay in the second term. b) The amount you pay in the first term.

NK E BANK A B E G A ANGU LANGUAG L 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 9 ANK GE BANK B B E E G G A A U U G A LAN LANG LANGUA LANGU

c) The amount you pay in the first month for three children.

NUMERAL SYSTEMS Natural numbers (1, 2, 3…) were introduced because people needed to count. They changed over time to adapt to new cultures and times. Prehistoric civilisations already had counting systems. For example, they used their fingers, cut notches into sticks, put beads on strings… As society evolved, people had to work with larger numbers and needed a more practical system. This is how different numeral systems emerged in different cultures.

This Palaeolithic man has written the number 47. What is the value of each symbol?

A numeral system is the set of symbols and rules used to represent numbers.

ã The Egyptian numeral system Ancient Egyptians used the following symbols:

This is the number 1 333 331.

100

1 000

10 000

100 000

1 000 000

stick

hobble

rope

flower

finger

frog

person

The way they wrote numbers was simple. They added the necessary symbols until they had the number they wanted. Numeral systems, like the Egyptian, that add symbols together until they get the desired value are called additive systems.

ã The Mayan numeral system The Mayan people lived in modern-day Guatemala and the south of Mexico. Before Columbus arrived in the Americas, the Mayans only used three symbols to write numbers: (0) (0)

0 1 5

2 7

3 8

4 9

(1) (1)

(5) (5)

Look at the diagram on the left. It shows how numbers under 20 were written using an additive system. These numbers are refered to as the first level. Larger numbers were written using the same symbols, but adding new levels. Each time they added a new level, the value of the symbols was multiplied by 20.

10 11 12 13 14

Second level (× 20) →

15 16 17 18 19

First level (× 1) →

20 20

21 21

27 27

36 36

40 40

100 100

137 137

The Mayan numeral system also has a characteristic of the positional numeral system because the value of the symbols changes depending on their level. The Mayan numeral system was partly additive and partly positional. 10

Unit 1

ã The decimal numeral system Nowadays, we use the decimal numeral system. It has ten symbols or figures (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) and follows these rules:

Remember We can break down a number into units and its different place values: 27 473

• We write the symbols in different levels called place values: units, tens, hundreds…

2 T Th → 20 000 7 Th → 7 000 4 H → 400 7 T → 70 3 U → + 3 27 473

For example:

• Ten units of one level form one single unit of the following level. • The value of a figure depends on its place value. The decimal system is positional. M

H Th

T Th

↓

4 000 000 U

↓

4 000 U

The value of 4 changes according to its place value. eas

id Consolidating

1 Think about the decimal numeral system.

a) How many tens are there in 3 thousands? b) How many hundreds are there in one ten of thousands?

Help H Th T Th

H × 10

c) How many hundreds are there in 5 one million units?

1 Th = 100 T

× 10

Let’s practise! 1 Write the following numbers using the Egyptian

numeral system: 19, 65, 34 120 and 2 523 083.

2 The following symbols are used in an additive system:

1 5 10 100 Write the following numbers using this system: 7, 12, 84 and 126. 3 Translate the following Mayan numbers into the

decimal system:

5 Complete the following in your notebook:

a) 500 T = … H = … Th b) 3 000 H = … Th = … T Th c) 6 Th = … H = … T d) 8 H Th = … T Th = … T 6 True or false?

a) If you move figures to a different place value, the value of the number they represent changes. b) If you add a zero to the right of a number, its value becomes ten times greater. c) If you add a zero to the left of a number, its value becomes ten times lower. d) Half a thousand equals 5 tens.

4 Look at the Mayan numbers below. Add four elements

to the left of the series. Now, add four to the right.

e) One thousand thousands equals one million. 7 A number contains five figures that add up to 5. If

you change the place value of the units to thousands, the total increases by 999. What is the number?

LARGE NUMBERS Many numbers contain more than nine figures. For example, there are 7 000 000 000 people on the Earth, 3 153 600 000 seconds in a century and 9 460 800 000 000 kilometres in a light year.

F ocu s on Eng lish The English words billions, trillions, quadrillions… are false friends. They do not mean the same in Spanish as in English. For example: 1 billion ↔ A 1 followed by 9 zeros. 1 trillion ↔ A 1 followed by 12 zeros.

hundreds

tens

units

The universe was formed thirteen thousand eight hundred million years ago.

millions

thousands

An uncommon name for a thousand millions (1 000 000 000) is a milliard. Sometimes, the prefix giga is also used. For example: 1 000 000 000 bytes = 1 gigabyte

billón

…

Interesting fact

thousands of millions

The decimal numeral system allows us to write numbers that contain as many figures as we want. The table below shows the place value for some numbers with more than 9 figures:

A young person’s brain contains around one hundred thousand millions of neurons.

The volume of the Earth is approximately one billón cubic kilometres.

• One million ↔ A 1 followed by 6 zeros. • One billón ↔ A 1 followed by 12 zeros. • One trillón ↔ A 1 followed by 18 zeros.

Let’s practise! 1 Read the first paragraph on this page. Then, write the

following numbers in words:

a) The number of people on Earth. b) The number of seconds in a century. c) The number of kilometres in a light year. 2 Write the following numbers in figures:

a) Twenty-eight million three hundred and fifty thousands. b) One hundred and forty-three millions. c) Two thousand seven hundred millions. d) Sixteen gigas. e) One and a half billón.

3 Copy and complete in your notebook:

a) One thousand thousands = one ... b) One thousand millions = one ... c) One million of millions = one ... 4 The human body contains between ten and seventy

million of millions cells. Express both amounts in billones.

5 How do you say the number that is written as a 1

followed by 30 zeros?

6 Scientists estimate that there are three cuatrillones

kilograms of water in our seas and oceans. What do you think a cuatrillón is?

Unit 1

ROUNDING NATURAL NUMBERS When a number has many figures, it is difficult to remember and makes calculations more difficult. We often replace it with an approximate value ended in zeros. That number is more manageable. For example: There are 31 853 000 €500 notes in circulation in Spain.

F ocu s on Eng lish In English, we write the € symbol before the amount (not after it like in Spanish). However, when we say the amount, we say Euro after it. For example: €500 ↔ Five hundred Euros.

Approximately, how many thousand million Euros is this?

more unit to the following figure.

ideas Consolidating

1 Use the diagram below to help you round the number 384 523 to the

nearest hundreds of thousands; to the nearest tens of thousands; and to the nearest thousands. hundreds of thousands tens of thousands thousands

3 83 348 854 425 532 23 3

+1 +1 +1 H8Th ≥85≥85≥ 5

= = =

... ...0...0 0 0 0 0 0

T Th4 <45<45< 5

3 83 348 854 425 532 23 3 +1 +1 +1 Th

... ...0... 0 0 0 0 0

Help

If we round the number 52 722: – to the nearest tens of thousands → 50 000 – to the nearest thousands → 53 000

5 ≥55≥55≥ 5

... ... ...0 0 0 0 0

Let’s practise! 1 Round the following numbers to the nearest thousands:

a) 24 963

b) 7 280

c) 40 274

d) 99 834

2 Round these numbers to the nearest hundreds and to

the nearest tens of thousands: a) 530 298 b) 828 502

c) 359 481 d) 29 935 236

3 Look at the newspaper in the picture. Round the

number of tourists to the nearest millions and the amount they spent to the nearest thousands of millions. , In 2018 0 00 82 600 isited v s tourist . They Spain 89 678 spent Euros. million

4 Round the following numbers to the nearest millions:

a) 24 356 000

b) 36 905 000

c) 274 825 048

5 The sign below shows the price of a house in Euros.

FOR SALE 138 290 € €138 290

€138 000 €138 300 €140 000

a) Which of the three approximations is closest to the real value? b) Which approximation would you use in an informal conversation? 6 The town hall has a budget of €149 637 to refurbish

its sports centre. Which approximate number would you use to tell a friend about this? anayaeducacion.es Practise rounding numbers. 13

BASIC OPERATIONS WITH NATURAL NUMBERS Although you already know how to operate with natural numbers, this section is a revision of some concepts and properties.

ã Addition and its properties

SEATING CAPACITY: 25 342 seats Seats filled

Addition means to find the total value of a set of numbers.

East stands: 11 576

Look at the picture on the left. To find the total number of people at the football stadium, we add the numbers together:

West stands: 9 006

11 576 + 9 006 = 20 582 Addition follows these two properties: • Commutative property: The sum does not change if the order of the

sumands changes.

Commutative property

• Associative property: The way in which the values are grouped does not

34 + 16 = 16 + 34 50 50

affect the result.

(a + b) + c = a + (b + c)

Associative property

ã Subtraction and its relation to addition

(18 + 3) + 17 = 18 + (3 + 17) 21 + 17 18 + 20 38 38

Subtraction means to “take” one number from another number, to calculate what is left. In other words, to find the difference. For example, if we want to count the empty seats at the football stadium above, we subtract the number of full seats from the total number of seats: 25 342 – 20 582 = 4 760

Remember

As you can see 25 342 = 20 582 + 4 760. And 20 582 = 25 342 – 4 760.

25 342 ← Minuend (M )

– 20 582 ← Subtrahend (S )

a+b=b+a

M =S +D Relationship between addition and subtraction: M – S = D → * S=M –D

4 760 ← Difference (D )

Let’s practise! 1 Calculate:

3 Transform:

a) 254 + 78 + 136

b) 340 + 255 – 429

a) This addition into a subtraction: 48 + 12 = 60

c) 1 526 – 831 + 63

d) 1 350 – 1 107 – 58

b) This subtraction into an addition: 22 – 2 – 6 = 14

2 Calculate and check the answer:

Carmen bought a bag for €167, a coat for €235 and a scarf for €32. How much did she spend in total? a) She spent around €350. b) She spent around €450. c) She spent around €550.

4 If Alberto were 15 years older, he would still be

18 years younger than his uncle Thomas. His uncle Thomas is 51 years old. How old is Alberto?

5 If I only bought a washing machine, I would still have

€246. However, if I also bought a TV, I would be €204 short. Could you say how much is any of these items? anayaeducacion.es Mental arithmetic: addition and subtraction.

Unit 1

ã Multiplication and its properties A multiplication is basically a repeated addition of the same value. For example, if a ticket to the football game from the previous page costs €35, the total price for all 20 582 tickets purchased is: Mental arithmetic

35 + 35 + 35 + … + 35 = 35 · 20 582 = €720 370

16 × 55

20 582 times

Multiplication follows these three properties:

8 × 2 × 5 × 11 88 × 10

• Commutative property: The product does not change if we change the

order of the factors.

880 The associative property allows us to regroup expressions. The commutative property allows us to change their order.

a·b=b·a

• Associative property: The product of a multiplication is not affected by the

way in which we group the factors.

(a · b) · c = a · (b · c) • Distributive property: The product of a calculation does not change if we

anayaeducacion.es Mental arithmetic: multiplication.

remove the parentheses.

a · (b + c) = a · b + a · c

a · (b – c) = a · b – a · c

The following example will help you to understand the distributive property: On Thursday, a group of friends bought 7 tickets to the game. On Friday, they bought 3 more tickets. How much did all the tickets together cost?

35 · 7 + 35 · 3 = 35 · (7 + 3) 245 + 105 35 · 10 350 350

We can write the calculation in two different ways: price of 7 tickets + price of 3 tickets ↔ price of (7 + 3) tickets 35 · 7 + 35 · 3 = 35 · 10

Let’s practise! 6 Complete the following multiplications in your

notebook:

5 × 2 + 9 0 1 2 6 0

9 8 × 2 8 7 4 + 6 9 9 3 4

7 Remember that to multiply by 10, 100, 1 000… you

just have to add one, two, three… zeros to the end of the number. a) 19 · 10 d) 140 · 10

b) 12 · 100 e) 230 · 100

c) 15 · 1 000 f ) 460 · 1 000

8 Write a mathematical expression:

Multiplying a number by eight is the same as multiplying it by ten, then subtracting double the original number. Which property describes this?

9 Multiply mentally by 9 and 11. Use the examples to

help you.

• 23 · 9 = 23 · 10 – 23 = 230 – 23 = 207 • 23 · 11 = 23 · 10 + 23 = 230 + 23 = 253 a) 12 · 9

b) 25 · 9

c) 33 · 9

d) 12 · 11

e) 25 · 11

f ) 33 · 11

10 A wheel turns at 1 500 revolutions per minute. How

many times does it turn in fifteen minutes? How many times does it turn in an hour? How many times does it turn in an hour and a half?

11 A farmer has an orchard with 200 peach trees. He

estimates that each tree can fill seven boxes with five kilos of peaches in each one.

How much profit does he make if he sells all of the peaches at €2 per kilo?

4 BASIC OPERATIONS WITH NATURAL NUMBERS

ã Division Remember two of the ways in which we can solve divisions that are common in everyday life: • We use 5 625 cubic metres of water to water a park for 15 days. How many cubic metres do we use each day? WATER FOR DAILY USE

5 6 2 5 1 1 2 075 00

15 375

⎯→ 5 625 : 15 = 375 m3 each day

Division means sharing a value between several people or things in equal parts to work out how much each one receives. • We use 375 cubic metres of water to water the park each day. There are 5 625 cubic metres of water in a tank. How many days does the water in the tank last? 5 6 2 5 1 8 7 5 000

375 15

⎯→ 5 625 : 375 = 15 days

Division means splitting a whole thing into equal portions of a specific size to work out how many portions there are.

ã Exact division and integer division Exact division We pack 35 kg of oranges into 5 kg boxes. 35 5 0 7 We fill 7 boxes and there are no remaining oranges. Integer division We pack 38 kg of apples into 5 kg boxes. 38 5 3 7 We fill 7 boxes and there are 3 kg remaining.

anayaeducacion.es Mental arithmetic: division.

In the previous example, we used 5 625 cubic metres of water to water the park for exactly 15 days. We had no water left. 375 5 6 2 5 1 8 7 5 15 ⎯→ 5 625 = 375 · 15 000 We call this an exact division. However, if there were 5 700 cubic metres of water in the tank, there would be some extra water left, even after 15 days. 375 5 7 0 0 1 9 5 0 15 ⎯→ 5 700 = 375 · 15 + 75 075 We call this an integer division. There are two types of division depending on the value of the remainder: • Exact division (the remainder is zero).

D d ⎯→ The dividend is equal to the divisor multiplied by 0 q the quotient. D=d·q • Integer division (the remainder is not zero). D d ⎯→ The dividend is equal to the divisor multiplied by r q the quotient, plus the remainder.

D=d·q+r

Unit 1

ã A property of division Example

×7

32 0

Look at what happens if we multiply the dividend and the divisor by the same number:

8 4

To water 3 plants, we need 24 litres of water. What happens if we have twice the number of plants and twice the amount of water?

×7

224 56 00 4

24 litres

The quotient does not change.

48 litres

24 3 0 8

48 6 0 8

If we double the amount of water and the number of plants, the amount of water each plant receives does not change. In a division, if we multiply the dividend and divisor by the same number, the quotient does not change. Let’s practise! 12 Find the quotient and remainder of each division:

a) 96 : 13

b) 713 : 31

c) 5 309 : 7

d) 7 029 : 26

e) 49 896 : 162

f ) 80 391 : 629

b) 180 : 12

c) 300 : 12

d) 75 : 15

e) 90 : 15

f ) 180 : 15

g) 180 : 30

h) 240 : 30

i) 390 : 30

14 Copy and complete the diagrams below in your

notebook:

(36 : 12) : 3 :

1 000 12

divisor 38

b) The remainder is always lower than the divisor.

: 4

a) 60 : 12

53 15

a) The quotient must be greater than the divisor.

: 12

: 3

dividend 39 17 True or false?

13 Follow the example to divide mentally.

• 96

16 Find the missing value in each division:

36 : (12 : 3)

What have you noticed? 15 Solve the following divisions and compare your

results. Then, answer the question. a) (50 : 10) : 5

50 : (10 : 5)

b) (36 : 6) : 2

36 : (6 : 2)

Does division follow the associative property?

c) In an exact division, multiplying the dividend by 2 doubles the quotient. d) Multiplying the dividend and the divisor by 3 triples the quotient. e) Division follows the commutative property. 18 Solve the following problems without making notes:

a) 150 grams of salami are divided between three sandwiches. How many grams are there in each sandwich? b) How many minutes are there in 180 seconds? c) We have travelled 240 kilometres in three hours. How many kilometres did we travel each hour? d) We put 250 kg of apples in 10 kg boxes. How many boxes are there? 19 A farmer collects 1 274 eggs. He puts 30 eggs in each

tray, and 10 trays in each box. How many eggs are left over? How many trays are left over?

EXPRESSIONS WITH COMBINED OPERATIONS ã Order of operations to solve an expression

• 48 : 3 + 5 – 2 · 3 = 16 + 5 – 6 = = 21 – 6 = 15

When solving expressions with combined operations, you must remember the rules of mathematical notation. These rules help us to make sure that each expression has a unique meaning and solution.

• 48 : (3 + 5) – 2 · 3 = 48 : 8 – 6 = =6–6=0

Look at the order followed in the calculations below. Although each calculation contains the same values and operations, the answers are not the same.

• 48 : 3 + (5 – 2) · 3 = 16 + 3 · 3 = = 16 + 9 = 25 Why? Write any number that contains two figures, a b . Now, write the figures in the opposite order, b a . Add both numbers together, and divide your answer by the sum of the two figures, a + b.

( a b

+ b a ) : (a + b) = …? What is the answer? Why?

48 : 3 + 5 – 2 · 3

48 : (3 + 5) – 2 · 3

48 : 3 + (5 – 2) · 3

16 + 5 – 6

48 : 8 – 6

16 + 3 · 3

21 – 6

6–6

16 + 9

The order of combined operations is always: 1st Brackets. 2nd Multiplication and division. 3rd Addition and subtraction.

eas

id Consolidating

1 Complete each box in your notebook. Check that you have the right answers.

4 · 10 – 8 · 3 + 2 4 · (10 – 8) · 3 + 2 4 · 10 – (8 · 3 + 2) – + 2 4· ·3+2 – ( + 2) +2 +2 – 18

4 · 10 – 8 · (3 + 2)

– 8 ·

–

4 · (10 – 8) · (3 + 2) 4 · · 4 · 40

ã How to use a calculator Type this sequence into a calculator: 2 + 3 * 4 = It may seem strange, but different calculators give different answers. You could have got 20 or 14.

{∫“≠} → The calculator performs each operation in the order it was entered. (2 + 3) · 4 = 5 · 4 = 20

{∫‘¢} → The calculator performs the multiplication first. It follows the correct order of operations.

2 + 3 · 4 = 2 + 12 = 14 As you can see, not all calculators work in the same way. Find out which method your calculator uses. Remember this when you use it. 18

Unit 1

ideas Consolidating

2 Copy and complete in your notebook. Check your answers with a calculator.

Make sure you enter the operations in the correct order. 40 – 12 : 4 + 2 · 3 40 – +

(40 – 12) : 4 + 2 · 3 :4 +

40 ≤ 12 / 4 µ 2 * 3 ≤Ñ→ {∫∫∫∫∫∫¢«}

Help

≤ → Add screen value to memory.

µ → S ubtract screen

value from memory.

ecover memory Ñ→ R value.

40 - 12 =/ 4 ≤ 2 * 3 ≤Ñ → {∫∫∫∫∫∫‘«}

Let’s practise! 1 Follow the examples to solve the following operations:

5 Problem solved

• 12 – 2 · 4 = 12 – 8 = 4

This month, an employee worked for 7 hours on 12 days earning his normal pay rate. He also worked 9 hours on 5 days. During 6 of those hours, he earned his normal pay rate. For the remaining 3 hours, he earned the night-time rate. How many hours did he work in total this month?

• (17 – 5) : 3 = 12 : 3 = 4 a) 8 + 5 · 2

b) 15 – 10 : 5

c) 4 · 6 – 13

d) (15 – 3) : 4

e) (8 + 2) · 3

f ) 18 : (10 – 4)

2 Calculate the following in your head. Compare your

We can solve this by writing it in two different ways:

results.

a) 2 + 3 · 4

(2 + 3) · 4

b) 6 – 2 · 3

(6 – 2) · 3

c) 18 – 10 : 2

(18 – 10) : 2

d) 24 : 6 + 2

24 : (6 + 2)

normal rate

12 · 7 + 5 · 6 + 5 · 3 = 84 + 30 + 15 = 129 12 days

a) 2 · (7 – 3) – 5

b) 3 · (10 – 7) + 4

c) 4 + (7 – 5) · 3

d) 18 – 4 · (5 – 2)

e) 8 – (9 + 6) : 3

f ) 22 : (7 + 4) + 3

4 Calculate. Write down the steps you followed. Check

your answers on the right. If they do not coincide, go over what you have done again. a) 6 · 4 – 2 · (12 – 7)

⎯→ 14

b) 3 · 8 – 8 : 4 – 4 · 5

⎯→ 2

c) 21 : (3 + 4) + 6

⎯→ 9

d) 26 – 5 · (2 + 3) + 6

⎯→ 7

e) (14 + 12) : 2 – 4 · 3

⎯→ 1

f ) 2 · (6 + 4) – 3 · (5 – 2) ⎯→ 11 g) 30 – 6 · (13 – 4 · 2)

⎯→ 0

h) 3 · [13 – 3 · (5 – 2)]

⎯→ 12

5 days

12 · 7 + 5 · (6 + 3) = 84 + 5 · 9 = 84 + 45 = 129 Answer: In total, he worked 129 hours.

3 Follow the example to calculate:

• 4 · (7 – 5) – 3 = 4 · 2 – 3 = 8 – 3 = 5

night-time rate

Write the following statements as mathematical expressions and solve them. a) A van carries 8 boxes of bananas, 20 boxes of oranges and 6 boxes of apples. Each box of bananas weighs 15 kg. Each box of oranges and each box of apples weighs 8 kg. How many kilograms of fruit does the van carry? b) A supermarket orders 20 crates of full-fat milk, 15 crates of skimmed milk and 10 crates of semiskimmed milk. Each crate holds 6 one-litre bottles. How many bottles did the supermarket order? c) There are 15 tables, 55 chairs and 12 stools in a cafe. How many legs are there in total? (hint: each stool has 3 legs). d) A farmer packs 1 500 eggs into boxes that hold 10 eggs, another 1 500 eggs into boxes that hold 6 eggs, and 300 free-range eggs into boxes that hold 6 eggs. How many boxes does he fill? 19

io”, nk “Portfol . source ba ur portfolio acion.es re yo du te ae ea ay cr to In the an ce on how an id gu d you will fin

ROBLEM P D N A S E IS C R EXE

Numeral systems 1

b) 235

c) 2 130

The table below contains data on vegetable consumption in Spain in 2016:

Write the following in Roman numerals: a) 87

Write the following numbers using the Egyptian additive system: a) 48

Translate the following Egyptian numbers into the decimal numeral system: a

b) 425

c) 2 600

weight

d) 54 528

Write the number fifty-seven using at least three different numeral systems. How many figures are there in a billón? And in a trillón? How many zeros are there in each number? a) One million is equal to one thousand hundreds.

c) One thousand times a million is equal to one giga.

Rounding 8

Copy and complete the table in your notebook: rounding number

2 830 554 19 270 000 399 675 000

to the nearest to the nearest hundreds of thousands million

4 369 449

6 195 054

vegetables and potatoes

3 626 510

5 214 031

total

7 995 959

11 409 085

Here are some hotel room numbers: 401; 235; 724; 231. a) Which one is at the end of the corridor? b) Which one is on the top floor? c) Which ones are on the same floor?

Do you remember how car number plates are ordered? Have a look at these plates:

e) One billón is equal to one million of millions. Star A is five light years away. Star B is five billones kilometres away. Which star is the furthest?

fresh fruit

What can numbers tell us?

d) One hundred gigas are equal to one billón. 7

(thousands of €)

Copy the table in your notebook, but rounding the figures to the nearest million tonnes and to the nearest hundred million Euros.

True or false? b) One hundred millions is equal to one thousand hundreds of thousands.

value

(tonnes)

3948 FBG

3894 FBG

4389 GFB

a) Which plate is the oldest? And the newest? b) Which is the plate number directly after the red one? And the previous one? c) How many plates were made between the red and the green ones? d) How many cars had the same letters as the blue plate after it?

Unit 1

Operations

Copy and complete in your notebook: 8 6 6

Addition and substraction 14

Calculate: a) 6 070 + 893 + 527 c) 831 – 392 – 76

b) 651 + 283 – 459 d) 1 648 – 725 – 263

Copy, calculate and complete in your notebook: a) 48 + … = 163 b) … + 256 = 359 c) 628 – … = 199 d) … – 284 = 196 Mental arithmetic. a) 5 + 7 – 3 – 4 c) 10 – 6 + 3 – 7 e) 12 + 13 + 8 – 23

b) 18 – 4 – 5 – 6 d) 8 + 5 – 4 – 3 – 5 f ) 40 – 18 – 12 – 6

Calculate: a) 47 – (35 – 28) b) 52 – (36 – 27) c) 128 – (86 – 45 – 12) d) 237 – (152 + 48 – 14) e) 348 – (148 – 86 + 29) f ) 235 – (340 – 152 – 84)

b) 128 · 10 e) 85 · 100 h) 134 · 1 000

14 5

Copy and complete in your notebook: a) 123 · … = 5 904

b) … · 86 = 1 548

c) … : 57 = 26

d) 1 862 : … = 133

Mental arithmetic. a) 3 · (10 : 5)

b) (4 · 6) : 8

c) 20 : (2 · 5)

d) (30 : 5) · 3

e) 10 : (40 : 8)

f ) (40 : 8) : 5

Calculate the following in your head. Remember that dividing by 5 is the same as dividing by 10, and then multiplying by 2. : 5

• 90

Calculate: a) 5 – [7 – (2 + 3)] b) 3 + [8 – (4 + 3)] c) 2 + [6 + (13 – 7)] d) 7 – [12 – (2 + 5)] e) 20 – [15 – (11 – 9)] f ) 15 – [17 – (8 + 4)] Check your answers: a) 3; b) 4; c) 14; d) 2; e) 7; f ) 10

Multiply: a) 16 · 10 d) 17 · 100 g) 22 · 1 000

8 2 9 7 6

: 10

· 2

a) 60 : 5

b) 80 : 5

c) 120 : 5

d) 140 : 5

e) 170 : 5

f ) 200 : 5

g) 210 : 5

h) 340 : 5

i) 420 : 5

Copy and complete in your notebook:

6 · (8 + 2) = 6 · 8 + 6 · 2 = 60

… = 5 · 9 – 5 · 6 = ....

(10 – 8) · 4 =

…

= ....

… = 7 · 12 – 2 · 12 = ....

Which property did you use?

Multiplication and division 19

5 3

c) 60 · 10 f ) 120 · 100 i) 140 · 1 000

Calculate the quotient and remainder: a) 2 647 : 8 b) 1 345 : 29 c) 9 045 : 45 d) 7 482 : 174 e) 7 971 : 2 657 f ) 27 178 : 254

Mental arithmetic.

a) One barrel can hold 5 litres of water. How many barrels can you fill with 100 litres of water? b) One kilo of almonds costs €12. How much do you pay for 5 kilos? c) There are 24 cans of soft drink in a box. How many cans are there in 10 boxes? d) It costs €360 to replace all four tyres on a car. How much does each tyre cost? 21

EXERCISES AND PROBLEMS 27

True or false? a) Multiplying a number by three gives the same result as doubling the number and adding it to the original number. b) Three times fifteen is the same as fifteen times three. c) Multiplying by ten is the same as multiplying by five twice. d) Multiplying by ten is the same as multiplying by five, then by two. e) The commutative property only applies to even numbers.

d) 3 · (2 + 5) – 13 h) 2 · 3 + 5 · (13 – 4 · 3)

Check your answers: a) 2; b) 11; c) 47; d) 8; e) 9; f ) 14; g) 9; h) 11 Interpret, describe and express 33

Match each statement to two of the mathematical expressions below: I. There are 50 people on a bus. 16 people get off the bus at the first stop and 4 people get on. II. There are 50 students in a music class. Today, 4 are absent without an explanation. Another 16 are absent because they are at a concert. III. Ernest buys a t-shirt for €16 and a hat for €4. He pays with a €50 note.

b) 2 · 4 + 6 d) 5 · 7 – 5 f ) 5 + 6 : 3 h) 18 – 7 · 2

IV. Last night, 50 clients slept at a hotel. This morning, 16 more people arrived and 4 people left. a) 50 – 16 – 4

b) 50 – 16 + 4

c) 50 – (16 + 4)

d) 50 – (16 – 4) e) 50 + (16 – 4) f ) 50 + 16 – 4 b) 8 : 4 + 7 – 3 d) 10 – 12 : 6 – 4 f ) 8 + 10 : 5 – 10 h) 11 – 2 – 9 : 3 j) 15 : 3 + 7 + 4 : 2 l) 12 : 4 – 1 – 6 : 3 n) 9 : 3 + 8 : 4 – 7 : 7 p) 18 : 2 – 12 : 3 – 6 : 2

Use the numbers 9, 3 and 1 in an operation to produce the values shown on the scales: a

c) 5 · (11 – 3) + 7 g) 3 · 5 – 3 · (10 – 4 · 2)

Calculate: a) 8 + 7 – 3 · 4 c) 15 – 2 · 3 – 5 e) 22 – 6 · 3 + 5 g) 36 – 8 · 4 – 1 i) 4 · 7 – 13 – 2 · 6 k) 5 · 4 + 12 – 6 · 4 m) 5 · 6 – 4 · 7 + 2 · 5 o) 8 · 8 – 4 · 6 – 5 · 8

b) 5 + 3 · (8 – 6)

e) 2 · (7 + 5) – 3 · (9 – 4) f ) 4 · (7 – 5) + 3 · (9 – 7)

Investigate: In division, if we multiply the dividend and divisor by the same number, the quotient remains the same. What happens to the remainder?

Combined operations 29 Calculate: a) 2 · (4 + 6) c) 8 : (7 – 5) e) (5 + 6) · 4 g) (19 – 7) : 2

Calculate: a) 30 – 4 · (5 + 2)

Which equation or equations solve the following problem? This morning, a supermarket sold 24 kg of apples for €2/kg, 12 melons for €4 each and 13 pineapples for €2 each. How much profit did it make from the sale of this fruit?

a) 24 · 12 + 4 · 13 + 2

b) 24 · 2 + 12 · 4 + 13 · 2

c) (24 + 13) · 2 + 12 · 4

d) (24 + 13 + 2) · (2 + 4)

Read the problem and look at the solution. Can you explain what each operation shows? There are horses, cows and chickens at a farm. In total, there are 714 legs, 168 horns and 137 beaks. How many horses are there at the farm? Solution: 1. 168 : 2 = 84

2. 84 · 4 = 336

3. 137 · 2 = 274

4. 336 + 274 = 610

5. 714 – 610 = 104

6. 104 : 4 = 26

Unit 1

Problem solving 36

Problem solved

Explain the steps followed for each operation. Explain the result.

A food retailer pays €2 000 for 150 bags of potatoes. Each bag contains 30 kg of potatoes. He examines the potatoes and decides to throw 300 kg away. The remaining are packed into 5 kg bags, and each bag is sold for €4. What is the retailer’s profit?

Last week, Maria sent 40 text messages. She sent five to her brother, Peter. She sent to her parents three more messages than she sent to Peter. She sent the remaining to her friends’ group chat. How many messages did she send to her friends?

Clara was paid €28 to deliver 7 piles of publicity. How much would she be paid if she delivers an extra pile?

Every day, a baker bakes five trays of muffins. Each tray contains three dozen muffins. The bakery is closed on Mondays. How many muffins does she bake in a week?

— Kilos bought (150 bags × 30 kg): 150 · 30 = 4 500 kg — Kilos packed (he throws away 300 kg): 4 500 – 300 = 4 200 kg — 5 kg bags obtained: 4 200 : 5 = 840 bags — Income, in Euros, for the sale of 840 bags at €4 each: 840 · 4 = €3 360 — Profit (€3 360 income minus €2 000 expenses): 3 360 – 2 000 = €1 360 Answer: The retailer makes €1 360. 37

A jam factory makes 250 kg of plum jam. The jam is put into 200 g jars. During the process, they throw out 17 jars of jam due to defects. How many valid jars of jam did it produce?

It took 14 months to build house A. Work began 4 months after the builders started to work on house B, which took them 15 months to build. If they finished building house A in June, when did they finish building house B?

On a farm, there are twice as many cows as there are horses. In total, there are 36 heads. How many cows are there? How many horses?

A van transports 15 boxes of orange soft drinks and 12 boxes of lemon soft drinks. Each box contains 24 cans. How many cans are there in total?

Jonathan is the father in the Smith family. He earns 1 940 dollars each month. He earns 720 dollars more than his son, John. He earns 880 dollars more than his daughter, Cathy. He earns 280 dollars less than his wife, Catherine. How much money does the Smith family earn each month in total?

Rosie is two years older than her brother Julian and two years younger than her brother Albert. Their mother is 42 years old and her age is equal to the total ages of all three siblings. How old is each sibling?

A gardener has 50 trays for planting seeds. He plants 100 seeds in each tray. However, an average of 20 seeds in each tray are spoiled. Approximately, how many plants will the gardener obtain? On a supermarket shelf there are 7 boxes of cans plus 4 cans out of box. Each box contains 6 cans. An employee puts 12 more boxes of cans on the shelf. How many cans are there in total?

A train carrying goods travels at 55 km/h. On the track next to it there is a passenger train travelling at 105 km/h. What is the distance between the two trains after half an hour? A car and a motorbike leave a cafe at the same time in the same direction. The car travels at 90 km/h and the motorbike travels at 100 km/h. What is the distance between them after an hour and a half? 23

EXERCISES AND PROBLEMS 50

A van transports 27 boxes of soft drinks. Each box contains 24 bottles. There is a traffic accident and the boxes tip over causing 311 bottles to break. Does the van still have at least half of its original load?

It is possible to form four different numbers of three figures only using zeros and ones. 1stª 1.

There are 54 tourists on a bus travelling to the airport, but the bus breaks down. The tourists need to get to the airport quickly, before their plane leaves. The group leader decides to put the tourists into taxis that can transport four people each. How many taxis do they need?

3rdª 3. 1 0 1 0

22.ndª

111 110 101 100

How many four figure numbers are there that only contain zeros and ones? And five figure numbers? 60

There are five starters, three main courses and two deserts on a menu. How many different meals can you make if you choose one dish from each category?

61 52

Marta has saved €162 because she wants to buy a skateboard that costs €199. If she saves €10 each week, how many weeks until she can buy the scooter?

Anthony, Beatrice, Claire and David go to the cinema. They have four seats next to each other. In how many different ways can they sit?

A car manufacturer makes 15 660 cars between January, February and March. On average, how many cars does it make each day?

1st 2nd 3rd 4th

The local tourist industry employs 12 845 people. Three in every five employees are women. How many female employees are there?

There are 450 students at a school. Two in every five students study a second language. One in three of these students studies German. How many students study a second language? How many study German?

First, let’s solve an easier problem: In how many different ways can they sit if Anthony sits in the 1st seat? A B C D

B D C

C B D

C D B

D B C

D C B

A company that organises events orders 150 dozen roses from a florist. The florist has 40 boxes of 25 roses in stock at the moment. How many more boxes of 25 roses does the florist need?

A farmer has an orchard with 140 peach trees. Based on his experience, he expects each tree to produce an average of 35 kg of peaches. He stores the peaches in 10 kg boxes that he sells for €20 each. How much profit does he make?

Marta, Julian and Rosie go shopping. Marta spends €30 more than Julian and €40 less than Rosie. In total, they spend €208. How much does each friend spend? You have a large pile of 50, 20 and 10 cent coins. In how many different ways can you add them together to make 1 Euro? Explain your answer.

Victoria has a farm with ducks and geese. Today, she sold 21 of her animals for €350. She sold twice as many ducks as geese. One goose costs three times the price of a duck. How much does one duck cost? How much does one goose cost?

Unit 1

A car travels 2 km in 78 seconds. The speed limit is 90 km/h. Does the driver exceed the speed limit? Explain why. A rectangular field measures 150 m × 300 m. A farmer wants to plant trees in it. She plants them in rows, parallel to the fences surrounding the field, with a distance of 5 metres between each row. The distance between the first row and the fence is also 5 metres. How many trees can she plant?

‘+’ problems 69

A number contains four figures that add up to 4. If you swap the ones for hundreds, the total increases by 99. What is the number? There is more than one possible answer.

Gemma and Fred live in the same building and go to the same school. When Gemma goes to school alone, she takes 20 min to get to class. It takes Fred 30 min to do the same journey, along the same route. Today, Gemma leaves the building five minutes after Fred. How long does it take her to catch him?

Draw a grid to help you. For example: 20

20 15

The chart below shows the colour of the 30 690 cars produced in a quarter.

We know that of the students in the first year of secondary school: — 44 eat in the cafeteria, 58 take the school bus and 47 participate in extracurricular activities. — 24 students eat in the cafeteria and participate in extracurricular activities. — 23 eat in the cafeteria and take the school bus; 25 take the school bus and participate in extracurricular activities. — 11 students do all three of these things and 17 students do not do any. How many students are there in total?

GREY WHITE GREEN BLUE

RED OTHERS

How many red cars were produced? 67

Larry conducts a survey on holiday destinations for people from a city. He obtains the following: — 56 % of the people surveyed went to the beach. — 47 % went to a village. — 23 % went to both destinations. What percentage of people surveyed did not go to the beach or to a village?

Mary added the first seven natural numbers together as follows: 1+ 2 + 3 + 4 + 5 + 6 + 7 8 · 7 = 56 + 7 + 6 + 5 + 4 + 3 + 2 + 14 56 : 2 = 28 8+8+8+8+8+8+8 Can you add the numbers 1 to 100 together?

Draw a Venn diagram like this one to help you.

YEAR 1 CAFETERIA

SCHOOL BUS

EXTRA ACT.

Four friends decide to weigh themselves in pairs. They do this in all possible combinations and write down the results in a random order: 83 kg - 87 kg - 91 kg - 80 kg - 84 kg - 88 kg The tallest child weighs 46 kg. How much does each child weigh separately?

A motorcycling competition is held at the Laguna Seca track. The green motorcycle takes 1 minute and 46 seconds to complete each lap but it started badly. The red one started well, but it takes 1 minute and 48 seconds to complete each lap. The red motorcycle passes the start line and the green one crosses 3 seconds later. There is still a long time left until the end of the race. How long will it take the green motorcycle to overtake the red one? 25

MATHS WORKSH READ AND LEARN

The history of counting Would we be able to live in the modern-day world without numbers? The birth and development of civilizations was always hand in hand with the birth and development of numbers. Over time, complex tools were developed to represent and make calculations with numbers. Hands were used as early calculators. Later, people developed methods for working with larger numbers. For example, they used piles of stones, beads on strings, abacuses… People invented mechanical calculators in more recent times. Electronic calculators and computers are even more modern. They are capable of managing enormous numbers and can do calculations almost instantly! Multiplication in ancient Egypt Ancient Egyptians multiplied numbers by two, then by two again and again, until they reached the number they wanted. For example, this is how they would calculate 23 × 18. They made two columns of numbers following these rules: — In the first column, they doubled the number 1 until they reached the first number in the calculation without going over it. In this case, without going over 23. ←• 1 ⎯→ ←• 2 ⎯→

— In the second column, they doubled the second number, 18 in our example, the same number of times as they doubled the number 1 in the first column.

18 → 36 →

— Then, they added all the numbers they needed from the first column together to make 23: 1 + 2 + 4 + 16 = 23

←• 4 ⎯→ 72 → 8 144

←• 16 ⎯→ 288 → 414 ← → 23

• Use this method to solve the following multiplications: a) 17 × 41

b) 41 × 17

INVESTIGATE Many cultures used abacuses throughout history. One of the most effective was the Chinese abacus. The following images show how people used it to calculate 326 + 15:

+15

• Work out how they moved the beads to do this calculation. • Then, draw diagrams to show how you would use the abacus for the following calculations: a) 211 + 42 26

b) 131 – 6

Unit 1

PRACTICE MAKES PERFECT! Try it out • Copy the diagram below in your notebook. Write the numbers 1 to 9 in each box. You can only use each number once. Each set of three numbers along each line should add up to 15.

• How many three figure numbers can you make using only the numbers 1, 2 and 3?

Complete the table below to help you: squares

corners

remaining corners

triangles

Not possible

SELF-ASSESSMENT

0 1 5

• In April 2018, the world population was 7 601 767 200.

10 11 12 13 14 15 16 17 18 19

decimal

528

Which systems are additive? Which systems are positional? What is the difference? 2 Copy and fill in the blanks in your notebook:

a) 18 ·

= 180

c) 4 000 :

= 40

· 100 = 27 000

: 10 = 38

3 Copy and complete in your notebook:

a) 154 ·

= 462

c) 30 275 :

= 35

· 125 + 8

4 Solve the following combined operations:

a) 12 + 3 · 5 – 2

b) 19 – 5 · (10 – 7) + 4 · 7

c) 7 · 3 – 4 · 2 + 2

d) 10 · [7 · 5 – (4 + 6 · 3)]

5 There are 60 seats in a cafe. There are three times

more chairs than stools. How many chairs are there? How many stools?

Commitment

90 km/h in the opposite direction.

What is the distance between them after ten minutes?

: 27 = 98

d) 1 508 =

6 Read the following statements:

• Brazil has a surface area of eight million five hundred and fourteen thousand eight hundred and seventy-seven square kilometres.

numeral systems

mayan

anayaeducacion.es Answer key.

1 Complete the following table in your notebook:

egyptian

8 A beekeeper has 187 hives. She collects two harvests

Watch the video for target 13.3. 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, Danny Latimer, R. Oakes, J. Roe, Hannah Peat, Karen Piper, Deborah Spencer, Denise Suárez, Begoña Fuente Larrazabal, Sara Gascón Martín, C. Ordóñez, Ana Villarroya, J. Colera Jiménez, Ignacio Gaztelu Albero and Ramón Colera Cañas. The following people have worked on this book:

Editorial team: Federica Cocco, Beatriz Fuentes and Sara Gascón Martín INCLUSION AND ANTI-DISCRIMINATION ADVISOR: Víctor Díez Design, technical drawings and maps: Miguel Ángel Castillejos, Patricia G. Serrano, Juan Carlos Quignon Illustrations: Alberto Hoyos Layout: DiScript and Isabel Román Corrections: Miguel Ángel Alonso and Sergio Borbolla Graphic edition: Olga Sayans Translation: Montero Language Services Proofreading: Imanol Miqueleiz Legaz unit opening pages: Deborah Spencer Photographs: Age Fotostock (Christoph Papsch, Hans Blossey, Stefano Baldini, Stuart Gregory, txking, Zoonar/Eder Hans), Archivo Anaya (Candel, C., Cosano, P., García Pelayo, Á., Leiva, Á., Martín, J.A., Martinez, C., Moreno, C., Ortiz, J., Padura, S., Peñuela-Py, E., Pérez-Uz, B., Redondo, M., Ruiz, J.B., Sánchez, J., Valls, R.), Dreamstime (Evgeny Karandaev, Oleksii Terpugov, Stocksolutions, Vampy1), Getty Images (FotografiaBasica, Jetlinerimages, 4X-image), iStock/ Getty Images (airdone, Albert_Karimov, alex-mit, Alexey Kabanov, AnatolyM, bazilfoto, f9photos, gedzun, jjmm888, MarcelC, mawielobob, MilaDrumeva, Milkos, Nadezhda1906, neamov, Nerthuz, Photitos2016, pkazmierczak, ronstik, scanrail, Seregraff, talevr, Ирина Мещерякова, 3sbworld), NASA/JPL-Caltech, 123RF(alphaspirit, Anna Om, Artem Demidenko, Eric Isselee, Evgeniy Skripnichenko, floralset, macrovector, Michael Schmeling, oly5, Ramzi Hachicho, rawpixel, robuart, Timmary, Volha Bilevich,1xpert).

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.

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a c r ce

DEMO

MAtemáticas

1 ESO

Building

Blocks

e rca c e d

MAtemáticas ESO Índice

....................................................... 2 ..... ..... ..... ..... ..... . s. ale tur na ros 1. Los núme ...........8 ............................................................ 2. Potencias y raíces.................... ............14 ............................................................ ..... ..... ..... ..... ..... ..... ad lid ibi vis Di 3. ...20 ............................................................ ..... ..... ..... ..... os ter en ros me nú s 4. Lo .......................................................26 ..... ..... ..... ..... s... ale cim de ros me 5. Los nú ......... 32 ............................................................. 6. El sistema métrico decimal. .........38 ............................................................ 7. Las fracciones........................... ....44 ............................................................ ..... . . es on cci fra con s ne cio era 8. Op .........................................................50 ..... . . jes nta rce po y ad alid ion 9. Proporc ...........56 ............................................................ 10. Álgebra...................................... ..........64 ............................................................ ..... ..... ..... ..... . . los gu án y s cta Re 11. ......72 ............................................................ ..... ..... ..... s... ica étr om ge s ura 12. Fig ..................................................... 80 ..... ..... ..... ..... ..... ..... . . os etr rím pe y 13. Áreas ................................................84 ..... ..... ..... ..... ..... ..... s... ne cio fun 14. Gráficas de .......... 90 ............................................................ 15. Estadística................................ .... 98 ............................................................ ..... ..... ..... ..... ad ilid ab ob pr y zar 16. A

D iDA

los números naturales Números en el sistema maya

1 SISTEMAS DE NUMERACIÓN

A medida que la sociedad evolucionaba, se hizo necesario manejar cantidades

• Los números menores que 20 se escriben:

grandes y representarlas de una forma práctica. Así, aparecieron los sistemas de numeración.

0 1 5

El sistema de numeración egipcio

4 9

10 11 12 13 14

El sistema de numeración egipcio es un sistema aditivo porque para escribir un número, se van añadiendo (sumando) los símbolos necesarios hasta completar la cantidad.

15 16 17 18 19

• Los números mayores que 20 se escriben: x 20 →

100

1 000

10 000

100 000

palo

asa

cuerda

flor

dedo

rana

x 1 →

1 000 000 hombre20

20 36 20

100

137

21 100 27 137 36 40 21 27 36

40 40

El sistema de numeración maya El sistema de numeración maya es un sistema en parte aditivo y en parte posicional. Tiene solo tres símbolos que asumen valor diferente según el nivel en el que se encuentran. (Fíjate en el esquema ❶). (0) (0)

(1) (1)

(5) (5)

Valor posicional del 4 en 4 304

El sistema de numeración decimal

El sistema de numeración que utilizamos actualmente es el decimal.

• Consta de diez símbolos o cifras (0, 1, 2, 3, 4, 5, 6, 7, 8 y 9). Se definen órde-

nes de unidades: unidades, decenas, centenas…

• Diez unidades de un orden hacen una unidad del orden inmediato superior. • El valor de una cifra depende del lugar que ocupe (sistema de tipo posicio-

nal). (Fíjate en el esquema ❷).

↓

4 000 U

↓

La cifra 4, en el número 4 304, tiene diferente valor según el orden de unidades que ocupa.

Practica… 1 Responde verdadero o falso.

a) El sistema de numeración egipcio es aditivo porque importa la posición de los símbolos. b) El sistema decimal es posicional porque el valor de una cifra depende del lugar que ocupe. c) El sistema de numeración maya es posicional.

2 Copia y completa en tu cuaderno.

Un sistema de numeración es … si al ir añadiéndose símbolos se suma su cantidad representada y es posicional si el valor de cada símbolo depende de la … que ocupe. Un sistema de numeración que combina ambas características sería el … .

100 100

Unidad 1

2 LOS NÚMEROS GRANDES

Observa

El sistema de numeración decimal permite representar cantidades tan grandes como deseemos. • Un millón ↔ Un 1 seguido de 6 ceros. • Un billón ↔ Un 1 seguido de 12 ceros.

1 0

1 0 0

3 0 0

8 0 0

0 0 0

millares

millones

billones

miles de millones

• Un trillón ↔ Un 1 seguido de 18 ceros.

0 0 0

El universo se originó hace trece mil ochocientos millones de años. El cerebro de una persona joven tiene unos cien mil millones de neuronas. La Tierra tiene un volumen aproximado de un billón de kilómetros cúbicos.

Las palabras inglesas billions, trillions, quadrillions… son falsos amigos porque no significan lo mismo que en castellano. Por ejemplo: 1 billion ↔ Un 1 seguido de 9 ceros. 1 trillion ↔ Un 1 seguido de 12 ceros.

Practica... 1 Copia en tu cuaderno y

completa. a) Mil millares son un … b) Mil millones son un … c) Un millón de millones es un …

2 Escribe con cifras.

a) Veintiocho millones trescientos cincuenta mil. b) Ciento cuarenta y tres millones. c) Dos mil setecientos millones.

3 APROXIMACIÓN DE NÚMEROS NATURALES

Cuando un número tiene muchas cifras, es difícil de recordar e incómodo para operar. Por eso, lo solemos sustituir por otro más manejable de valor aproximado, terminado en ceros. La forma más frecuente de realizar aproximaciones es el redondeo. Veamos un ejemplo de cómo redondear.

Practica... 1 Lee esta noticia y aproxi-

Aproximación de 52 722 a los millares. Se sustituyen por ceros todas las cifras a la derecha de dicho orden.

… 000.

+1 Si la primera cifra sustituida es mayor o igual que … 53 000. cinco, se suma una unidad a la cifra anterior.

52 722 7≥5

ma el número de turistas a los millones y el gasto a los miles de millones. visitaron En 2018a 82 600 000 Españ ristas que 8 tu n 89 67 gastaroillones m ros. de eu

Propiedades de la suma

4 OPERACIONES BÁSICAS CON NÚMEROS

Propiedad conmutativa

La suma y sus propiedades

34 + 16 = 16 + 34

Recuerda que sumar es unir, juntar, añadir. Por ejemplo, si queremos saber el número de personas que hay en el campo de fútbol que se ve en el margen, deberemos hacer una suma. La suma cumple las siguientes propiedades: • Propiedad conmutativa: La suma no varía al cambiar el orden de los sumandos.

Propiedad asociativa (18 + 3) + 17 = 18 + (3 + 17)

a+b=b+a

21 + 17 38

18 + 20

• Propiedad asociativa: El resultado de la suma es independiente de la forma en

que se agrupen los sumandos.

(a + b) + c = a + (b + c)

AFORO: 25342 localidades Localidades ocupadas Gradas este: 11576 Gradas oeste: 9006

(Fíjate en el esquema ❶).

La resta y sus relaciones con la suma Recuerda que restar es quitar, suprimir, hallar lo que falta o lo que sobra; es decir, calcular la diferencia. Por ejemplo, para saber cuántas localidades vacías hay en el partido de la imagen de la derecha, antes, tenemos que realizar una resta: 25 342 – 20 582 = 4 760 Observa que 25 342 = 20 582 + 4 760 y que 20 582 = 25 342 – 4 760.

Recuerda

25 342 ← minuendo (M )

– 20 582 ← sustraendo (S )

Relaciones entre la suma y la resta: M – S = D

4 760 ← diferencia (D )

M=S+D S=M–D

Practica... 1 Calcula.

a) 254 +78 + 136 c) 340 + 255 – 429

3 Transforma.

b) 1 526 – 831 + 63 d) 1 350 – 1 107 – 58

2 Estima la respuesta y compruébala después.

Carmen compra un bolso de 167 €, una gabardina de 235 € y un pañuelo de 32 €. ¿Cuánto se ha gastado? a) Se ha gastado alrededor de 350 €. b) Se ha gastado, más o menos, 450 €. c) Se ha gastado alrededor de 550 €.

a) Esta suma en una resta: 48 + 12 = 60 b) Esta resta en una suma: 22 – 2 – 6 = 14 4 Si Alberto tuviera 15 años más, aún sería 18 años

más joven que su tío Tomás, que tiene 51 años. ¿Cuál es la edad de Alberto?

5 Si comprara solo una lavadora, me sobrarían 246 €,

pero si comprara también un televisor, me faltarían 204 €. ¿Puedes decir el precio de alguno de estos artículos?

Unidad 1

La multiplicación y sus propiedades Recuerda que multiplicar es una forma abreviada de realizar una suma repetida de sumandos iguales. Por ejemplo, si una entrada para el partido de fútbol de la página anterior costaba 35 €, la recaudación por las 20 582 entradas vendidas sería: 35 + 35 + 35 + … + 35 = 35 · 20 582 = 720 370 €

20 582 veces

La multiplicación cumple las siguientes propiedades: • Propiedad conmutativa: El producto no varía al cambiar el orden de los factores.

a·b=b·a • Propiedad asociativa: El resultado de una multiplicación es independiente de

la En forma quede se amigos agrupeny los factores. unaenpeña amigas, compraron el jueves 7 entradas para el partido, y el viernes, 3 entradas más para los rezagados. ¿Cuál fue el coste de (a · b) · c = a · (b · c) las entradas? Podemos distributiva: calcular de dosElformas el coste las entradas: • Propiedad producto de undenúmero por una suma (o resta) es • gasto de 7 entradas de 3 entradas igual a la suma (o resta)+degasto los productos del número por cada sumando. • gasto de (7 + 3) entradas a · (b + c) = a · b + a · c a · (b – c) = a · b – a · c 35 · 7 + 35 · 3 = 35 · 10

Observa Propiedades de la multiplicación • La propiedad asociativa nos permite reagrupar los términos, y la propiedad conmutativa, cambiarlos de orden. 16 × 55

8 × 2 × 5 × 11 88 × 10 880 • Propiedad distributiva de la multiplicación 35 · 7 + 35 · 3 = 35 · (7 + 3)

245 + 105 350

35 · 10

350

Practica... 6 Copia y completa en tu cuaderno.

5 × 2 + 9 0 1 2 6 0

9 8 × 2 8 7 4 + 6 9 9 3 4

7 Expresa con una igualdad aritmética:

Multiplicar un número por ocho es lo mismo que multiplicarlo primero por diez y después restarle su doble. ¿Qué propiedad se aplica en esta igualdad?

8 Multiplica mentalmente por 9 y por 11 como se

hace en los ejemplos. • 23 · 9 = 23 · 10 – 23 = 230 – 23 = 207

• 23 · 11 = 23 · 10 + 23 = 230 + 23 = 253

a) 12 · 9 d) 12 · 11

b) 25 · 9 e) 25 · 11

c) 33 · 9 f ) 33 · 11

9 ¿Cuántas vueltas da en un cuarto de hora una rueda

que gira a razón de 1 500 revoluciones por minuto? ¿Y en una hora? ¿Y en hora y media?

La división

División entera y exacta

• Dividir es repartir un todo entre varios en partes iguales, para averiguar cuánto

• División entera (el resto es distinto de cero).

• Dividir es partir un todo en porciones iguales de un tamaño dado, para averi-

D d r c El dividendo es igual al divisor por el cociente más el resto:

le toca a cada uno.

guar cuántas porciones se obtienen.

La división: exacta y entera

Una división puede ser exacta o entera dependiendo del valor del resto. (Fíjate en el esquema ❷).

D=d·q+r • División exacta (el resto es

cero).

Una propiedad de la división

D d 0 c El dividendo es igual al divisor por el cociente (el resto es 0):

Observa lo que ocurre cuando en una división multiplicamos el dividendo y el divisor por el mismo número:

D=d·q

Para regar 3 arbustos, utilizamos 24 litros de agua. ¿Qué ocurre si tenemos el doble de arbustos y el doble de litros de agua? Al repartir el doble de litros entre el doble de arbustos, la cantidad que corresponde a cada uno no varía. 24 litros 48 litros 24

3 8

x2 x2

Observa Si en una división se multiplican el dividendo y el divisor por el mismo número, el cociente no varía.

Practica… 10 ¿Verdadero o falso?

12 Averigua el factor que falta en cada división:

a) El cociente debe ser mayor que el divisor. b) El resto es siempre menor que el divisor. c) Si es exacta, al multiplicar por dos el dividendo, el cociente se hace el doble. d) Al multiplicar por 3 el dividendo y el divisor, el cociente aumenta el triple. e) La división cumple la propiedad conmutativa. 11 Calcula y compara los resultados. Después, reflexio-

na y contesta.

a) (50 : 10) : 5

50 : (10 : 5)

b) (36 : 6) : 2

36 : (6 : 2)

¿Cumple la división la propiedad asociativa?

dividendo 39

53 15

1 000 12

divisor 38

13 Realiza en tu cuaderno las operaciones como indi-

can los colores.

(36 : 12) : 3

36 : (12 : 3)

¿Qué observas? 14 Un granjero recoge 1 274 huevos, los envasa en ban-

dejas de 30, y las bandejas, en cajas de 10.

¿Cuántos huevos quedan sin completar una bandeja? ¿Cuántas bandejas quedan sin completar una caja?

Unidad 1

5 EXPRESIONES CON OPERACIONES COMBINADAS Usa la calculadora

Orden en que han de hacerse las operaciones Al resolver expresiones con operaciones combinadas, debes tener en cuenta las normas del lenguaje matemático. Estas normas aseguran que cada expresión tenga un significado y una solución únicos. Observa el orden de actuación en las siguientes expresiones: los resultados son diferentes a pesar de estar formadas por los mismos números y operaciones. 48 : 3 + 5 – 2 · 3

48 : (3 + 5) – 2 · 3

48 : 3 + (5 – 2) · 3

16 + 5 – 6

48 : 8 – 6

16 + 3 · 3

21 – 6

6–6

16 + 9

Esto ocurre porque en cada expresión se está respetando una diferente prioridad jerárquica. Por eso, en las expresiones con operaciones combinadas, hemos de atender: • Primero, a los paréntesis. • Después, a las multiplicaciones y a las divisiones. • Por último, a las sumas y a las restas.

Introduce en la calculadora esta operación: 2+3.4= Aunque te parezca extraño, según la máquina que utilices puedes obtener en pantalla dos soluciones diferentes, 20 o 14.

{∫“≠} → La calculadora hace las

operaciones en el orden en que van entrando. (2 + 3) · 4 = 5 · 4 = 20 {∫‘¢} → La calculadora hace primero el producto. Es decir, respeta la prioridad jerárquica. 2 + 3 · 4 = 2 + 12 = 14 Como ves, no todas las calculadoras tienen la misma lógica interna. Averigua de cuál de los dos tipos es la tuya y tenlo en cuenta cuando la utilices.

Practica… 1 Completa en tu cuaderno cada casilla y comprueba

que obtienes el resultado que se indica. 4 · 10 – 8 · 3 + 2

– + 2

2 Escribe una expresión que resuelva cada enunciado

y calcula la solución.

a) Una furgoneta transporta 8 cajas de plátanos, 20 de naranjas y 6 de manzanas. Las cajas de plátanos pesan 15 kilos y las de naranjas y manzanas 8 kilos. ¿Cuántos kilos de fruta transporta la furgoneta? b) Un supermercado hace un pedido de 20 lotes de leche entera, 15 de leche desnatada y 10 de semidesnatada. Cada lote contiene seis cajas de litro. ¿Cuántas cajas van en el pedido?

c) En una cafetería hay 15 mesas, 55 sillas y 12 taburetes. ¿Cuántas patas hay en total? (nota: las mesas y las sillas son de 4 patas, y los taburetes, de 3). d) Un granjero envasa 1 500 huevos en cajas de 10 unidades, otros tantos, en cajas de 6 unidades, y una partida de 300 huevos de producción ecológica, también en cajas de 6 unidades. ¿Cuántas cajas ha llenado? 3 Observa el ejemplo y calcula.

4 · (7 – 5) – 3 = 4 · 2 – 3 = 8 – 3 = 5 a) 2 · (7 – 3) – 5

b) 5 · 2 + 4 · (7 – 5)

c) 3 · (10 – 7) + 4

d) 18 : 2 – 2 · (8 – 6)

e) 4 + (7 – 5) · 3

f ) 30 – 4 · (5 + 2)

g) 18 – 4 · (5 – 2)

h) 5 + 3 · (8 – 6)

i) 8 – (9 + 6) : 3

j) 5 · (11 – 3) + 7

k) 22 : (7 + 4) + 3

l) 3 · (2 + 5) – 13

Building blocks es un proyecto educativo de Anaya para Educación Secundaria. En la realización de esta obra han intervenido:

Equipo de edición: Virginia Álvarez, M.a Federica Cocco, Beatriz Robles y Elia Ureña (adaptación del libro Matemáticas 1.° ESO)

Asesoramiento para la inclusión y la no discriminación: Victor Díez Diseño, gráficos y cartografía: Miguel Ángel Castillejos, Miguel Ángel Diaz-Rullo, Marta Gómez, Patricia G. Serrano y Juan Carlos Quignón

Ilustración: Celia López Maquetación: DiScript Corrección: Miguel Ángel Alonso y Sergio Borbolla Edición gráfica: Olga Sayans Fotografía : Archivo Anaya (Cosano, p.)

compromiso Nuestras publicaciones mantienen el rigor en el uso y en la selección de los contenidos, en las imágenes y en el lenguaje, para cumplir con la no discriminación por razón de género, cultura u opinión. Grupo Anaya, considera la responsabilidad social y medioambiental uno de sus valores fundamentales. Por ello, se compromete a: · mejorar nuestros resultados en materia de medio ambiente, · reducir nuestras emisiones de carbono, · hacer uso de los recursos naturales de manera responsable y · eliminar cualquier impacto negativo de nuestra actividad en los bosques en peligro. Dichos compromisos, entre otros, hacen que el 100 % del papel utilizado en nuestros libros tenga el sello PEFC.

Importante: Las actividades propuestas en este libro deben ser realizadas en cuadernos u hojas sueltas; nunca en el propio libro. Los enlaces a las páginas web que aparecen en este libro han sido revisados en la fecha de su impresión. La editorial no se hace responsable de las modificaciones o las anulaciones que se produzcan en ellos con posterioridad a dicha fecha.