Mathematics 1 Teacher's Guide sample unit

DEMO

1

SECONDARY EDUCATION

Mathematics J. C oler a Ji m én ez, I . Gazt elu Al ber o , R . C o l er a Cañ as

T e a c h e r’s G u i d e

Building

Blocks

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: Beatriz Fuentes, Sara Gascón Martín and Joaquín Montón Design, technical drawings and maps: Miguel Ángel Díaz-Rullo, Patricia G. Serrano, Juan Carlos Quignon Illustrations: Alberto Hoyos Layout: DiScript and Isabel Román Graphic edition: Olga Sayans Translation: Montero Language Services and Robin Munby

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.

© 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.

INDEX Building blocks and the project keys ..............................

4

Our project ........................................................................................

6

Units

1. NATURAL NUMBERS .............................................................

8

2. POWERS AND ROOTS ........................................................... 22 3. DIVISIBILITY ............................................................................. 34 4. INTEGERS ................................................................................... 48 5. DECIMALS .................................................................................. 64 6. THE METRIC SYSTEM .......................................................... 76 7. FRACTIONS ................................................................................ 88 8. OPERATING WITH FRACTIONS ........................................ 100 9. PROPORTIONALITY AND PERCENTAGES .................... 112 10. ALGEBRA ..................................................................................... 124 11. LINES AND ANGLES ............................................................... 140 12. GEOMETRIC SHAPES ............................................................ 154 13. AREAS AND PERIMETRES .................................................. 180 14. GRAPHS OF FUNCTIONS ..................................................... 192 15. STATISTICS AND PROBABILITY ...................................... 208

g n i d l i u b

blocks

g n i n r a e l p ee k o t A project that is rooted in skills-based learning and the development of student commitment within the realities of available time.

Building Blocks is a new skills-based approach, with the utmost curricular rigour and a coherent and coordinated content sequence in the areas throughout the entire educational stage. It promotes linguistic communication skills, which are essential for assessing the knowledge that allows us to understand the world around us and develop social awareness.

Building Blocks offers the possibility of incorporating active methodologies, using cooperative learning and thinking strategies, promoting personal and social skills for emotion management and the development of entrepreneurship in a flexible way. It attends to academic and professional orientation, whilst embracing equality and inclusion, all within the framework of the Sustainable Development Goals that we must keep focused on over the coming years.

Project keys SDG

SDG Commitment Establishes

the Sustainable Development Goals as a framework for learning that prepares students towards committed citizenship.

Developing thinking

Proposes strategies to stimulate reflection, “learning how to think” and the development of critical and creative thinking habits.

Cooperative learning Offers

techniques to develop the skills that allowus to work together and efficiently in a diverse society.

Emotional education Offers emotional management tools to face the challenges of this complex educational stage.

Enterprising culture

romotes entrepreneurial thinking in its three P dimensions: personal, social and productive.

ICT Integrates the use of ICT in the learning process itself in a responsible, intelligent and ethical way.

Academic and professional orientation Helps students to get to know themselves, to understand the environment and to make decisions that allow them to confidently enter the labour market.

Assessment Incorporates strategies that allow students to participate in the assessment of their learning, analysing “what they have learned” and “how they have learned it”.

Linguistic Plan Develops communication skills both written and spoken and the tools needed to describe, present, instruct, comment, defend or refute ideas...

Our project THE STUDENT’S BOOK The books present content and activities that are adjusted to the curricular development of our community and that develop mathematical reasoning based mainly on solving problems taken from everyday situations. Using a skills-based methodology, it allows us to express, in a creative and innovative way, our commitment to the Sustainable Development Goals, demonstrating how Mathematics contribute to the achievement of these objectives.

THE DIGITAL PROJECT FOR TECHERS THE web www.anayaeducacion.es The teacher’s website is a tool that facilitates and enriches teaching work.

With it, you can adapt the contents to the needs of the students through additional resources or reinforce the most relevant educational aspects: • Course plan, the teacher’s guide and project information.

•D iversity and inclusion, to meet the diversity of students’ motivations, interests and learning styles through: - Key concepts and additional worksheets.

- Extension worksheets and competence development worksheets.

- Tasks, workshops and other resources.

• Assessment, a core value of the project, is supported through: -E valuation and activity generator: an activity bank which you can use to design tests for each topic.

- Predesigned assessment tests and records.

- Specific documentation.

- Assessment tools.

• Resource bank, with a wide variety of digital resources such as: - GeoGebra activities - Video tutorials - Self-assessments - Glossaries - Key concepts - Learn by playing - Problem solving practice Language Bank and more.

•L anguage Bank: created to further develop linguistic skills, this new educational tool can be used both by the teacher and the language assistant.

1

SECONDA

AT RY EDUC

3

ION

SECONDA

tics Mathema Gaz telu éne z, I. as ra Jim ra Cañ J. C ole R. C ole

Alb ero

AT RY EDUC

ION

tics Mathemadaemic for Ac dies Stu

,

zál ez, eira Gon ª J. Oliv as éne z, M. R. C ole ra Cañ ra Jim , J. C ole Alb ero I. Gaz telu

Building

Blocks

Building

Blocks

Teacher’s Guide

1

g

Buildin

Mathematics

THE TEACHER’S GUIDE

1

SECONDARY EDUCATION

Blocks

learning to keep

www.anayaeducacion.es

J. C olera Jiménez, I. Gaztelu Albero, R. C olera Cañas

SECONDARY EDUCATION

T e a c h e r’s G u i d e

I S B N 978-84-698-7408-0

9

788469 874080

3100508

There is a teacher’s guide for each student book with the solutions to the activities, methodological guidelines, suggestions for applying the project keys, etc.

Mathematics

Building

Blocks

THE DIGITAL BOOK All Building Blocks books have a digital version.

You can download the entire book or download it by units, with all its resources or in its lighter version (without the largest digital resources), which allows offline and online use. The resources of each unit are grouped by type and allow direct access to the resource bank and the Teacher’s guide.

AND FOR THE S TUDENTS? Resource bank

at www.a nayaedu cacion.es • Resourc con: es for the project k • Subject eys. key conc epts.

• Resourc es by unit .

Digital b ook A digital version o f the textb be down ook that loaded in can full or by its resourc u nits; with es or a lig all hter vers the heavie ion (with st digital o u t resources) .

1 NATURAL NUMBERS Unit presentation Natural numbers do not seem to obey any human intellectual ‘construction’. They have always naturally appeared in all cultures to count, order, measure, and so on.

The unit starts by comparing some of the most well-known numeral systems. This enables us to show, in addition to the historical development of how numbers are represented, that the concept of natural numbers is the same for all of them regardless of how it is expressed.

After reviewing the structure of the decimal numeral system, and discussing its advantages compared to other numeral systems, we work on reading and writing numbers with nine or more figures. The students are also reminded how to round and its advantages.

We then review basic operations with natural numbers and some of their properties, paying special attention to division as this is where more errors and gaps are detected both conceptually and operationally with regards to the algorithm.

When reviewing the operations, we practise calculations and prioritise problem solving. This ensures students review and improve concept construction.

Lastly, we build on solving expressions with brackets and combined operations.

There are three types of contents in this unit: • Theoretical aspects: – Numeral systems. The decimal numeral system.

– Properties of operations and their advantages when calculating.

• Calculation: – Algorithms for operations.

– Expressions with brackets and combined operations.

– Mental arithmetic.

• Using a calculator and knowledge of basic techniques.

Basic knowledge • Structure of the decimal numeral system.

• Reading and writing large numbers.

• Rounding.

• Mental and written arithmetic with the four operations.

• Basic use of the calculator.

• Solving simple expressions with combined operations.

• Solving problems with one and two operations.

Task preparation • Finding information about different numeral systems (ancient civilisations, binary system used for computer languages, etc.).

• Reviewing how to work with the four operations (detecting gaps).

• Showing the different types of calculators.

• Looking back at general strategies and methods for solving problems and describing the solving processes.

8

KC: key competences, CLC: competence in linguistic communication, CMST: competence in mathematics, science and technnology, DC: digital competence, LL: learning to learn, SCC: social and civic competence, SIE: sense of initiative and entrepreneurship and CAE: cultural awareness and expression.

Contents and competencies Unit contents

Core competencies

Opening page • Different ways of writing the same number • Different multiplication methods • Combined operations

CLC CMST

1. Numeral systems • The Egyptian numeral system • The Mayan numeral system • The decimal numeral system

CLC CMST DC LL SCC CAE

2. Large numbers

CLC CMST LL

3. Rounding natural numbers

CLC CMST SCC

4. Basic operations with natural numbers • Addition and its properties • Subtraction and its relation to addition • Multiplication and its properties • Division • Exact division and integer division • A property of division

CMST LL

5. Expressions with combined operations • Order of operations to solve an expression • How to use a calculator

CLC CMST DC LL SCC

Final pages • Exercises and problems • Maths Workshop • Self-assessment

CLC CMST DC LL SCC SIE CAE

Developing thinking Technique: Think and share with a partner.

It is beneficial for students to think about what they believe the answer is and to listen to their classmates’ ideas to compare opinions.

Once they have thought about it, we ask them to share their answers with the classmate sitting next to them, explaining how they reached their answer.

1

PROJECT KEYS

NATURAL NUMBERS

SDG commitment • Target 11.c.

• Target 13.3.

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

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.

Linguistic Plan • Skills: Listening and Reading (receptive skills).

Speaking and Writing (productive skills)

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

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

6

Cooperative learning Technique: • Pencils in the middle Enterprising culture • Productivity (productivity dimension): my project • Initiative (productivity dimension): I support changes ICT • Worksheets on Language Bank • Interactive activities • Mental arithmetic • Answer key for the self-assessment Assessment • Exercises and problems • Preparation of the portfolio

5

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

3

1

12

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:

1 In the title of the text you can find the concept of numeral systems. Try

a) 208 × 34

to define it using your own words.

b) 453 × 26

Combined operations

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

reminder of the value of each letter:

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

registration fee and €15 per month.

1

5

10

50

100

500

Match the following mathematical expressions to the descriptions below:

1 000

(20 + 15) · 3

20 + 15 · 3

15 · 3

a) The amount you pay in the second term.

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.

b) The amount you pay in the first term.

BANK GE BANK LANGUA LANGUAGE BANK GE BANK GE BANK LANGUA LANGUAGE NK LANGUA BANK 9 BANK GE BA GE BANK LANGUA LANGUAGE NK LANGUA LANGUAGE GE BANK LANGUAGE BALANGUAGE BANK LANGUA c) The amount you pay in the first month for three children.

8

Developing thinking Technique: • Think and share with a partner

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

Different multiplication methods

Natural numbers Starting the unit In the Listening and Reading section the students will start reading and listening an introductory text about the different ways of expressing natural numbers. It also proposes that students think about the usefulness of numeral systems, how they are different and the role they have played in different cultures and ages.

It also presents a multiplication algorithm that is different to the ones that students already know about but which is similarly based on the decimal numeral system. Here we can take the opportunity to compare it with the algorithm used by the Egyptians (see page 26 of the student’s book) and show the relationship between the way numbers are represented and the advantages or disadvantages when operating with them.

Questions to detect preconceptions • Create a signs system to code any number lower than 50 (or 100…).

• Read and write numbers that have a maximum of eight figures.

• Calculate using basic operations.

• Compare very simple expressions varying the position of the brackets.

• Invent problems for a given operation.

Answer key 1 Open answer.

2 1 346 + 834 = 2 180. 2 180 = MMCLXXX 3 The number is 3 059.

MMMLX H Th T Th Th

first and third.

5 Open answer.

8 6 a) 208 × 34

H

T

U

4 The

3

2 4 4 0 0 3 2 2 0 6 0 0 2 0 8

0

0

7

b) 453 × 26

3

0

6 10 1 0 0 7 0 7 2 7

a) 15 · 3

b) 20 + 15 · 3

2

0 6 6 1 0 1 8 4 0 8 3 0 8 2 4

5

7

11 7 1 1 7 7 8

c) (20 + 15) · 3 9

1. Numeral systems Unit 1

1

NUMERAL SYSTEMS

ã The decimal numeral system

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.

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…

• 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.

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

For example: M

H Th

T Th

4

7

8

↓

1 Think about the decimal numeral system.

10

100

1 000

10 000

100 000

1 000 000

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.

5

6

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) →

U

4

↓

4U

H Th T Th

Th

H × 10

1

T

U

0

1 Th = 100 T

× 10

0

Let’s practise!

1 Write the following numbers using the Egyptian

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

1

5

10

100

5 Complete the following in your notebook:

Write the following numbers using this system: 7, 12, 84 and 126.

a) 500 T = … H = … Th

3 Translate the following Mayan numbers into the

b) 3 000 H = … Th = … T Th

decimal system:

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.

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

0

c) 6 Th = … H = … T

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:

2

b) How many hundreds are there in one ten of thousands? c) How many hundreds are there in 5 one million units?

2 The following symbols are used in an additive system:

ã The Mayan numeral system

0 1

T

3

Help

a) How many tens are there in 3 thousands?

1 stick

(0) (0)

H

4 4 000 U

ideas Consolidating

Ancient Egyptians used the following symbols:

This is the number 1 333 331.

Th

↓

4 000 000 U

The value of 4 changes according to its place value.

ã The Egyptian numeral system

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?

11

Suggested methodology Using different numeral systems that were developed in different periods and cultures will help students understand the continuous effort done to construct tools we use today without even thinking about the difficulties in the process, and that form part of our cultural heritage, which is continuously changing and is passed from one generation to the next.

We can also highlight the fact that each culture used a numeral system that met their needs. We cannot think of any situation in which a primitive human being, who was a hunter and gatherer, would have to handle number with more than seven figures. But we simply have to open any journal or scientific treatise to see that these are the same numbers that are essential for today’s society. By this, we mean that numeral systems were perfected as needs to number and calculate developed (trade, building, statistics, etc.). At the same time, each development has enabled new scientific fields to open up and has brought with it the emergence of new numerical requirements.

In order to fully appreciate the advantages of our decimal numeral system, we must compare it to other systems, especially additive systems. Show the difficulty of representing large numbers and decimal numbers with these systems, and also the difficulties of operating with them.

We recommend emphasising that the use of positional systems was a huge leap forward in the use of more concise symbols and their ability to express amounts. The importance of the late appearance of zero will also be noted. It was an abstract symbol used to fill a place where there was nothing, but at the same time, it was key for the development of these systems. Zero allowed different values to be assigned to the figures.

Answer key for ‘Consolidating ideas’ The activity included in this section aims to review the structure of the DNS and detect gaps in their understanding, providing support to overcome them.

1

a) 3 thousands are 300 tens. b) 1 ten of thousands are 100 hundreds. c) 5 one million units are 50 000 hundreds.

Answer key for 'Let's practise!'

1 19 = 65 = 34 120 = 2 523 083 = 2

7=

12 = 3

6

4 To

84 =

11

120

126 = 126

the left:

To the right:

5

a) 500 T = 50 H = 5 Th c) 6 Th = 60 H = 600 T

6 a) True 7

10

b) True

b) 3 000 H = 300 Th = 30 T Th d) 8 H Th = 80 T Th = 80 000 T c) False

40 001, so 41 000 – 40 001 = 999

d) False

e) True

2. Large numbers Unit 1

2

3

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.

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

1

units

tens

hundreds

thousands

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

1

3

8

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

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

more unit to the following figure.

ideas Consolidating

F ocus on English 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.

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

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

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

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

• One million ↔ A 1 followed by 6 zeros.

3 83 348 854 425 532 23 3

+1 +1 +1 H8Th ≥85≥85≥ 5

• One billón ↔ A 1 followed by 12 zeros.

= = =

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

• One trillón ↔ A 1 followed by 18 zeros.

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!

Let’s practise!

1 Read the first paragraph on this page. Then, write the

1 Round the following numbers to the nearest thousands:

3 Copy and complete in your notebook:

following numbers in words:

a) 24 963

a) One thousand thousands = one ... b) One thousand millions = one ... c) One million of millions = one ...

a) The number of people on Earth. b) The number of seconds in a century. c) The number of kilometres in a light year.

million of millions cells. Express both amounts in billones.

a) Twenty-eight million three hundred and fifty thousands.

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

b) One hundred and forty-three millions.

a) 530 298

b) 828 502

In 2018, 00 82 600 0 visited tourists They Spain.

89 678 spent Euros. million

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

e) One and a half billón.

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

number of tourists to the nearest millions and the amount they spent to the nearest thousands of millions.

6 Scientists estimate that there are three cuatrillones

d) Sixteen gigas.

4 Round the following numbers to the nearest millions:

d) 99 834

c) 359 481 d) 29 935 236

3 Look at the newspaper in the picture. Round the

followed by 30 zeros?

c) Two thousand seven hundred millions.

c) 40 274

the nearest tens of thousands:

4 The human body contains between ten and seventy

2 Write the following numbers in figures:

b) 7 280

2 Round these numbers to the nearest hundreds and to

€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.

12

13

Focus on English This Focus on English section focusses on the words 'billion' and 'trillion' as false friends.

Suggested methodology Large number (with six, nine, twelve and more figures) often appear in scientific, sociological, economic data and many more. This is why it is important to prepare and interpret messages related to resources that students already use.

Students must read and write numbers with many figures and work with the corresponding place values (millions, thousands of millions, billones, etc.) and their equivalents with ease.

In the next unit, they will learn to use the abbreviated notation for these place values with the help of powers of base ten.

We also recommend reinforcing the difference between our term billón and the term ‘billion’ which tends to appear in English texts and media and often leads to mistakes in translations. Counterintuitively, ‘billion’ is the same as a thousand millions (mil millones).

In an attempt to differentiate it from billón, and to have an equivalent term in translations, the new term milliard (mil millones) has been coined, although it is not often used.

Answer key for ‘Let's practise!’

1 a) Seven thousands of millions. b) Three thousand, one hundred fifty-three million six hundred thousands. c) Nine billones, four hundred and sixty thousand eight hundred millions.

2

a) 28 350 000 b) 143 000 000 c) 2 700 000 000 d) 16 000 000 000 e) 1 500 000 000 000

3

a) Million

4 Between 5

b) Thousand millions c) Thousand millions d) Billón

10 and 70 billones of cells.

Ten thousand billones.

6 A

1 followed by 24 zeros → one billón of billones.

3. Rounding natural numbers Unit 1

2

3

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.

hundreds

tens

units

3

8

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

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

1

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

more unit to the following figure.

ideas Consolidating

F ocus on English 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.

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

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.

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

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!

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?

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, 00 82 600 0 visited tourists 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.

12

13

ICT anayaeducacion.es In ‘My web resources’ there are interactive activities to practise this content.

Focus on English This Focus on English section focusses on the position of the Euro sign (€) in English.

Suggested methodology In addition to learning the meaning of the term rounding and become proficient at rounding amounts, students must get used to doing these operations to correctly express, remember or record data related to the information and answers to calculations they use daily.

When we are told on television that, for example, ‘the winners of 14 will collect 119 274 Euros’, we remember, and if necessary communicate, the information: ‘those of 14 will collect 120 000 Euros’. It is a different situation when one of the winners collects the prize.

Here the exact amount will be needed.

In order for the learning to be incorporated into the students’ competences, we can give them the activity of preparing a list of situations (like the one in the example) where rounding is suitable and effective (prices, budgets, statistical population data, economy, etc.).

Answer key for ‘Consolidating ideas’ The activity proposes rounding the same number to different place values.

The provided example, as well as a helpful diagram, aims to make the task easier and comprehensively strengthen the method. In addition, comparing the different cases will help create new connections and expound on the structure of numbers.

1

Hundreds of thousands: 400 000 Tens of thousands: 380 000 Thousands: 385 000 11

Answer key for 'Let's practise!'

1 a) 25 000 b) 7 000

c) 40 000

2

a) 530 298 → 500 000 and 530 000 c) 359 481 → 400 000 and 360 000

3

There were 83 000 000 millions of tourists, approximately.

They spent approximately 90 000 000 million Euros.

4 a) 5

24 000 000

a) 138 300

b) 37 000 000

d) 100 000

b) 828 502 → 800 000 and 830 000 d) 29 935 236 → 29 900 000 and 29 940 000

c) 275 000 000

b) 140 000

6 €150 000

4. Basic operations with natural numbers (I) Unit 1

4

BASIC OPERATIONS WITH NATURAL NUMBERS

ã Multiplication and its properties A multiplication is basically a repeated addition of the same value.

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

50

a+b=b+a

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

34 + 16 = 16 + 34

affect the result.

50

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

ã Subtraction and its relation to addition

(18 + 3) + 17 = 18 + (3 + 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 )

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

4 760 ← Difference (D )

Let’s practise!

3 Transform:

1 Calculate:

a) 254 + 78 + 136

b) 340 + 255 – 429

c) 1 526 – 831 + 63

d) 1 350 – 1 107 – 58

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.

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:

Associative property

21 + 17

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:

a) This addition into a subtraction: 48 + 12 = 60 b) This subtraction into an addition: 22 – 2 – 6 = 14 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?

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

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

350

35 · 7 + 35 · 3 = 35 · 10 Let’s practise!

6 Complete the following multiplications in your

notebook:

× 2

5

+ 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?

anayaeducacion.es Mental arithmetic: addition and subtraction.

14

15

ICT anayaeducacion.es In ‘My web resources’ there are interactive activities to practise mental arithmetic.

Suggested methodology Here we find a space to consolidate learning that started in previous years, which will serve as preparation to tackle operations with integers and fractions later on, where similar techniques will be used to those practised here.

We review the algorithms, as well as the properties and relations to addition and subtraction for two reasons: — Its automatic and spontaneous implementation to improve calculation.

— Its theoretical formalisation (expression with letters) so that students go beyond the specific example and apply them to all numbers.

Understanding properties and implementing them at a practical level must be done at these ages through experimentation and practise, rather than through analytical reasoning.

Therefore, the theoretical explanation will be given after comprehension, and it will be the last step of the learning process.

To support this practical implementation, it is a good idea to point out to students the advantages of applying the properties to make calculating products easier. This is especially important when developing mental arithmetic strategies, as shown in these examples: • The product of 35 × 12 can be made into a simpler one, 42 × 10, by combining the associative and commutative properties.

35 × 12 = (7 × 5) × (2 × 6) = 7 × (5 × 2) × 6 (1) = 7 × 10 × 6 = 7 × 6 × 10 = 42 × 10 (2) (1) Associative property (2) Commutative property • The product 125 × 23 is easier to calculate with the distributive property: 125 × 23 = 125 × (20 + 3) = 125 × 20 + 125 × 3 = 2 500 + 375 = 2 875 To expand upon this content, we propose taking out the common factor. This applies the distributive property in the opposite way to which it is usually found: a · b + a · c = a · (b + c) Answer key for 'Let's practise!'

1 a) 468 b) 166 2

The correct answer is b) 167 + 235 + 32 = €434.

3

a) 48 + 23 = 60 → 60 – 48 = 12 b) 22 – 2 – 6 = 14 → 14 + 2 + 6 = 22

4 51 5 6

– 18 – 15 = 18 years

The price of the TV is 204 + 246 = €450.

4 5 2 8 3 6 0

+ 9 0 1 2 6 0 ×

12

c) 758

9 5 8

× 7 3 2 8 7 4

+ 6 7 0 6 6 9 9 3 4

d) 185

7

a) 190 d) 1 400

b) 1 200 e) 23 000

c) 15 000 f ) 460 000

8

x · 8 = x · (10 – 2) = x · 10 – x · 2 We have applied the distributive property to this equality.

9

a) 12 · 9 = 12 · 10 – 12 = 120 – 12 = 108 b) 25 · 9 = 25 · 10 – 25 = 250 – 25 = 225 c) 33 · 9 = 33 · 10 – 33 = 330 – 33 = 297 d) 12 · 11 = 12 · 10 + 12 = 120 + 12 = 132 e) 25 · 11 = 25 · 10 + 25 = 250 + 25 = 275 f ) 33 · 11 = 33 · 10 + 33 = 330 + 33 = 363

10

In 15 minutes: 1 500 × 15 = 22 500 turns In an hour: 22 500 × 4 = 90 000 turns In one and a half hours: 22 500 × 6 = 135 000 turns

11

200 × 7 × 5 × 2 = €14 000

4. Basic operations with natural numbers (II) 4 BASIC OPERATIONS WITH NATURAL NUMBERS

Unit 1

ã Division

ã A property of 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

15 375

5625 112 075 00

⎯→

5 625 : 15 = 375 m3 each day

Example

×7

32 0

8 4

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

24 litres

224 56 00 4

24 0

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.

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.

16

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.

5625 1875 000

375 15

⎯→

5 625 = 375 · 15 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.

5700 1950 075

375 15

⎯→

We call this an integer division.

12 Find the quotient and remainder of each division:

a) 96 : 13

b) 713 : 31

c) 5 309 : 7

e) 49 896 : 162

f ) 80 391 : 629

: 12

• 96 :3

D 0

d q

⎯→ The dividend is equal to the divisor multiplied by 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

53 15

1 000 12

divisor 38

b) The remainder is always lower than the divisor.

:4

a) 60 : 12

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

(36 : 12) : 3

dividend 39

a) The quotient must be greater than the divisor.

8

32

16 Find the missing value in each division:

17 True or false?

13 Follow the example to divide mentally.

36 : (12 : 3)

:

:

There are two types of division depending on the value of the remainder: • Exact division (the remainder is zero).

6 8

Let’s practise!

notebook:

5 700 = 375 · 15 + 75

48 0

In a division, if we multiply the dividend and divisor by the same number, the quotient does not change.

d) 7 029 : 26

ã Exact division and integer division

3 8

If we double the amount of water and the number of plants, the amount of water each plant receives does not change.

• 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?

5625 1875 000

48 litres

The quotient does not change.

Division means sharing a value between several people or things in equal parts to work out how much each one receives.

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?

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?

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?

17

ICT anayaeducacion.es In ‘My web resources’ there are interactive activities to practise mental arithmetic.

Suggested methodology Students should already know how to use the division algorithm, but we will use this section to detect any possible gaps in their learning that would prevent them from acquiring subsequent content.

The division concepts will be reviewed using activities in suitable contexts (problem solving): — Division as sharing: this means finding out how many elements correspond to each part when a whole is divided into a specific number of equal parts.

— Division as splitting: this means finding out how many parts of a specific size can be made from the elements of a whole. This concept requires special attention as it is harder to grasp.

Relations between the terms of exact and integer division will be consolidated through checking and applying them to specific situations (for example, division test).

The section ends with an important property of division: what happens if we multiply the dividend and the divisor by the same number? Students can assimilate it through contextualised and simple examples. It will also be important to answer the question: what happens with the remainder? Applying this property will be fundamental to justify the division algorithms with decimal divisors, and will link to other content, such as the equivalence and simplification of fractions.

Answer key for ‘Let’s practise!’

12 a) c = 7; r = 5 b) c = 23; r = 0 d) c = 270; r = 9 e) c = 308; r = 0 13

a) 60 : 3 = 20 : 4 = 5 c) 300 : 3 = 100 : 4 = 25 e) 90 : 3 = 30 : 5 = 6 g) 180 : 10 = 18 : 3 = 6 i) 390 : 10 = 39 : 3 = 13

b) 180 : 3 = 60 : 4 = 15 d) 75 : 3 = 25 : 5 = 5 f ) 180 : 3 = 60 : 5 = 12 h) 240 : 10 = 24 : 3 = 8 36 : (12 : 3)

(36 : 12) : 3

14

c) c = 758; r = 3 f ) c = 127; r = 508

3 : 3

36 : 4

1

9

We can see that the division does not follow the associative property.

13

15

a) (50 : 10) : 5 = 5 : 5 = 1 b) (36 : 6) : 2 = 6 : 2 = 3 50 : (10 : 5) = 50 : 2 = 25 36 : (6 : 2) = 36 : 3 = 12 The division does not follow the commutative property.

16

53 · 15 + 39 = 834

(1 000 – 12) : 38 = 988 : 38 = 26

17

a) False d) False

b) True e) False

c) True

18

a) 50 g

b) 3 minutes

c) 80 km/h

19

1 274 : 30 → quotient = 42 and reminder = 14. There are 14 eggs left over.

42 : 10 → quotient = 4 and reminder = 2. There are two trays left over.

d) 25 boxes

5. Expressions with combined operations 5

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

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

15

0

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.

b + b a ) : (a + b) = …?

What is the answer? Why?

2nd Multiplication and division.

3rd

Addition and subtraction.

+2 +2

4 · (10 – 8) · 3 + 2 4·

·3+2 +2 26

4 · 10 – (8 · 3 + 2) – (

4 · 10 – 8 · (3 + 2) – 8 ·

+ 2)

–

–

14

0

4 · (10 – 8) · (3 + 2) 4 · 4

· · 40

ã How to use a calculator

40 –

+

:4

+

order of operations.

2 + 3 · 4 = 2 + 12 = 14

value from memory.

Ñ → Recover memory value.

40 - 12 =/ 4 ≤ 2 * 3 ≤Ñ → {∫∫∫∫∫∫‘«}

1 Follow the examples to solve the following operations:

5 Problem solved

• 12 – 2 · 4 = 12 – 8 = 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)

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?

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)

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

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

⎯→ 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

night-time rate

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

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:

a) 6 · 4 – 2 · (12 – 7)

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.

µ → Subtract screen

+ +

40 ≤ 12 / 4 µ 2 * 3 ≤Ñ→ {∫∫∫∫∫∫¢«}

your answers on the right. If they do not coincide, go over what you have done again.

(2 + 3) · 4 = 5 · 4 = 20

memory.

Let’s practise!

It may seem strange, but different calculators give different answers. You could have got 20 or 14.

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

Help

≤ → Add screen value to

(40 – 12) : 4 + 2 · 3

Type this sequence into a calculator: 2 + 3 * 4 =

{∫“≠} → The calculator performs each operation in the order it was entered.

18

40 – 12 : 4 + 2 · 3

results.

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

18

Make sure you enter the operations in the correct order.

2 Calculate the following in your head. Compare your

ideas Consolidating

–

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

• (17 – 5) : 3 = 12 : 3 = 4 The order of combined operations is always: 1st Brackets.

(a

4 · 10 – 8 · 3 + 2

Unit 1

ideas Consolidating

6

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

Cooperative learning Technique: Pencils in the middle.

Make groups of four students to do exercise 6. Each member of the team is given a section and asked to put their pencil in the middle of the table. Taking turns, the students read their sections and propose an answer. The rest of the group gives their opinion and they discuss how to solve it.

When they are sure of their answer, everyone in the group takes their pencil and answers the question without talking.

Suggested methodology Like any other language, mathematical language requires sequenced learning, which is checked through practice and that takes time.

Interpreting and producing arithmetic expressions with combined operations and brackets is not obvious for students. On the other hand, experience tells us that we must pay special attention to this aspect to ensure that learning errors that would affect subsequent learning do not occur.

In order to analyse different expressions and compare their differences, we recommend using diagrams that highlight their structure, as shown in the examples.

After calculating the value of an expression it is important for students to get used to expressing all the steps through successive horizontal equalities. Here we must look out for writing errors (these lead to equalities with partial answers by writing, for example 4 · (8 – 6) · 3 = 4 · 2 = 8 · 3 = 24).

Analysing the behaviour of different calculators when doing combined operations is also interesting. By showing students two calculators (one which follows the correct order of operation, and another simpler one that performs each operation as it is entered), they will be surprised to find that the same sequence of keys gives a different result in each: Calculator that respects order of operation: 4 + 6 × 3 → 22 Calculator that performs each operation in the order it is entered: 4 + 6 × 3 → 30 We reach the conclusion that to ensure we use a calculator correctly, we must have indepth knowledge about it and take into account how it works.

Answer key for ‘Consolidating ideas’ Exercise 1 gives different expressions with the same numbers and operations, but with different answers. The guided solution aims to help students internalise the role of brackets and the order of operations. Once the activity is completed, it can be complemented by horizontally writing out each process using successive equalities.

1

4 · 10 – 8 · 3 + 2

4 · (10 – 8) · 3 + 2

4 · 10 – (8 · 3 + 2)

40 – 24 + 2

4· 2 ·3+2

40 – ( 24 + 2)

16 + 2

24 + 2

40 – 26

18

26

14

4 · 10 – 8 · (3 + 2) 4 · (10 – 8) · (3 + 2) 40

14

– 8· 5

4 · 2

·

40 – 40

4 · 10

0

40

5

2

40 – 12 : 4 + 2 · 3 40 – 3 + 6

(40 – 12) : 4 + 2 · 3

28 : 4 +

37 + 6

7 + 6

43

13

6

Answer key for ‘Let’s practise!’

Exercises 5 and 6 aim to bring together the abstract nature of arithmetic expression and the daily life of students by giving them meaning in recognisable contexts that they can relate to.

1

a) 8 + 5 · 2 = 8 + 10 = 18 b) 15 – 10 : 5 = 15 – 2 = 13 c) 4 · 6 – 13 = 24 – 13 = 11 d) (15 – 3) : 4 = 12 : 4 = 3 e) (8 + 2) · 3 = 10 · 3 = 30 f ) 18 : (10 – 4) = 18 : 6 = 2

2

a) 14 and 20 b) 0 and 12 c) 13 and 4 d) 6 and 3 Comparing the answers shows that the brackets change the value of the expression.

3

a) 2 · (7 – 3) – 5 = 2 · 4 – 5 = 8 – 5 = 3 b) 3 · (10 – 7) + 4 = 3 · 3 + 4 = 9 + 4 = 13 c) 4 + (7 – 5) · 3 = 4 + 2 · 3 = 4 + 6 = 10 d) 18 – 4 · (5 – 2) = 18 – 4 · 3 = 18 – 12 = 6 e) 8 – (9 + 6) : 3 = 8 – 15 : 3 = 8 – 5 = 3 f ) 22 : (7 + 4) + 3 = 22 : 11 + 3 = 2 + 3 = 5 g) 5 · 2 + 4 · (7 – 5) = 10 + 4 · 2 = 10 + 8 = 18 h) 18 : 2 – 2 · (8 – 6) = 9 – 2 · 2 = 9 – 4 = 5

4 a) 6

· 4 – 2 · (12 – 7) = 24 – 2 · 5 = 24 – 10 = 14 b) 3 · 8 – 8 : 4 – 4 · 5 = 24 – 2 – 20 = 22 – 20 = 2 c) 21 : (3 + 4) + 6 = 21 : 7 + 6 = 3 + 6 = 9 d) 26 – 5 · (2 + 3) + 6 = 26 – 5 · 5 + 6 = 26 – 25 + 6 = 1 + 6 = 7 e) (14 + 12) : 2 – 4 · 3 = 26 : 2 – 12 = 13 – 12 = 1 f ) 2 · (6 + 4) – 3 · (5 – 2) = 2 · 10 – 3 · 3 = 20 – 9 = 11 g) 30 – 6 · (13 – 4 · 2) = 30 – 6 · (13 – 8) = 30 – 6 · 5 = 30 – 30 = 0 h) 3 · [13 – 3 · (5 – 2)] = 3 · [13 – 3 · 3] = 3 · [13 – 9] = 3 · 4 = 12

5

Problem solved.

6 a) 8

· 15 + (20 + 6) · 8 = 120 + 26 · 8 = 120 + 208 = 328 kilos b) (20 + 15 + 10) · 6 = 45 · 6 = 270 boxes c) (15 + 55) · 4 + 12 · 3 = 70 · 4 + 36 = 280 + 36 = 316 legs d) 1 500 : 10 + 1 500 : 6 + 300 : 6 = 150 + 250 + 50 = 450 boxes

15

Exercises and problems ”, bank “Portfolio acion.es resourcecreate your portfolio.

In the anayaedu on how to guidance you will find

EXERCISES AND

Unit 1

PROBLEMS

Numeral systems 1

9

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

b

d

c

2

10

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

3

b) 235

c) 2 130

11

Write the following in Roman numerals: a) 87

b) 425

c) 2 600

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

5

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

(tonnes)

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

Copy and complete the table in your notebook: rounding number

2 830 554 19 270 000 399 675 000

20

to the nearest to the nearest hundreds of thousands million

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 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)

4 369 449

vegetables and potatoes

3 626 510

5 214 031

total

7 995 959

11 409 085

6 195 054

18

What can numbers tell us?

12

d) One hundred gigas are equal to one billón.

Rounding

15

17

13

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

3948 FBG

E

3894 FBG

E

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?

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

20

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

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

9

5 7 6

22

23

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)

24

d) (30 : 5) · 3 f ) (40 : 8) : 5

Numeral systems 1 a) 57 b) 234

:5

9

2

a)

f ) 200 : 5

h) 340 : 5

i) 420 : 5

3

a) 87 = LXXXVII b) 425 = CDXXV c) 2 600 = MMDC d) 54 528 = LIVDXXVIII

c) 120 : 5

Copy and complete in your notebook:

(10 – 8) · 4 =

…

= ....

… = 7 · 12 – 2 · 12 = ....

Which property did you use?

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

c)

18 b) 80 : 5 e) 170 : 5

g) 210 : 5

… = 5 · 9 – 5 · 6 = ....

26

b)

d) 3 430 000

·2

a) 60 : 5 d) 140 : 5

6 · (8 + 2) = 6 · 8 + 6 · 2 = 60

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

c) 2 540

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

25

b) (4 · 6) : 8

c) 20 : (2 · 5) e) 10 : (40 : 8)

• 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

14

8 2

3 6

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

Multiplication and division 19

5 6

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

16

Copy and complete in your notebook: 8

Addition and substraction 14

(thousands of €)

fresh fruit

21

Operations

value

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

a) One million is equal to one thousand hundreds.

8

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

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

7

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.

d) 54 528

4

6

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?

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

Assessment anayaeducacion.es In ‘My web resources’ there are documents to prepare a portfolio.

SDG commitment Watch the video for target 11.c before doing exercise 9.

Start a class discussion on actions we can do to make cities more sustainable.

4 Decimal:

57 Roman: LVII Egyptian:

5

One billón → 1 000 000 000 000 → 13 figures, 12 zeros.

One trillón → 1 000 000 000 000 000 000 → 19 figures, 18 zeros.

6 a) False

d) False 7

Linguistic Plan Skill: Written expression (argumentative text).

In exercise 10, ask students to justify their answer in writing.

b) True e) True

c) True

1 light year → 9 and a half billones kilometres.

9 500 000 000 000 Star A → 5 años luz ≈ 45 billones de kilómetros.

Star B → 5 billones kilometres.

Star A is further away than star B.

Rounding 8

rounding to the nearest hundreds of thousands

to the nearest million

2 830 554

2 800 000

3 000 000

19 270 000

19 300 000

19 000 000

399 675 000

399 700 000

400 000 000

number

9 10

Approximately 20 million inhabitants.

The third option is the closest. But it does not say it is a rounded figure.

The first option is not as accurate as the third, but says that it is a rounded figure.

11

weight rounded to million tonnes

value rounded to hundreds of millions of Euros

fresh fruit

4 000 000

6 000 000 000

vegetables and potatoes

4 000 000

5 000 000 000

total

8 000 000

11 000 000 000

What can numbers tell us?

12 a) 235 b) 724 13

16

c) 235 and 231

a) The oldest is 3948 FBG. The newest is 4389 GFB. b) The blue one comes after the red one, and the green comes before it. c) There are 54 number plates. d) The last number plate with the same letters would be 9999 GFB. 9 999 – 4 389 = 5 610 5 610 cars have number plates with the same letters.

”, bank “Portfolio acion.es resourcecreate your portfolio.

In the anayaedu on how to guidance you will find

EXERCISES AND

1

9

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

b

d

c

2

b) 235

c) 2 130

11

Write the following in Roman numerals: a) 87

b) 425

c) 2 600

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

5

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

c) One thousand times a million is equal to one giga. d) One hundred gigas are equal to one billón.

Star A is five light years away. Star B is five billones kilometres away. Which star is the furthest?

Rounding Copy and complete the table in your notebook: rounding

2 830 554 19 270 000 399 675 000

20

to the nearest to the nearest hundreds of thousands million

14

(thousands of €)

4 369 449

6 195 054

vegetables and potatoes

3 626 510

5 214 031

total

7 995 959

11 409 085

13

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

18

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

3948 FBG

E

3894 FBG

E

4389 GFB

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)

5

6

23

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

14

a) 7 490 c) 363

b) 475 d) 660

15

a) 48 + 115 = 163 c) 628 – 429 = 199

b) 103 + 256 = 359 d) 480 – 284 = 196

16

a) 5 d) 1

b) 3 e) 10

c) 0 f ) 4

17

a) 40 d) 51

b) 43 e) 257

c) 99 f ) 131

18

a) 5 – [7 – 5] = 5 – 2 = 3 b) 3 + [8 – 7] = 3 + 1 = 4 c) 2 + [6 + 6] = 2 + 12 = 14 d) 7 – [12 – 7] = 7 – 5 = 2 e) 20 – [15 – 2] = 20 – 13 = 7 f ) 15 – [17 – 12] = 15 – 5 = 10

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

a) 60 : 5

25

Addition and subtraction

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

• 90

9

18 ·2

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

26

19

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

20

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

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

9 7

22

24

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

14

8 2

3 6

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

16

17

5 6

15

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?

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?

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

Copy and complete in your notebook: 8

Addition and substraction

value

(tonnes) fresh fruit

21

Operations

What can numbers tell us?

12

e) One billón is equal to one million of millions.

number

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

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.

8

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

a) One million is equal to one thousand hundreds.

7

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

4

6

10

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

3

Operations

Unit 1

PROBLEMS

Numeral systems

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

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

Multiplication and division 19

a) 160 d) 1 700 g) 22 000

b) 1 280 e) 8 500 h) 134 000

c) 600 f ) 12 000 i) 140 000

20 a) q

= 330; r = 7 b) q = 46; r = 11 d) q = 43; r = 0 e) q = 3; r = 0

21

8 1 6 0 6 6 1 6

2 5 3 2

c) q = 201; r = 0 f ) q = 107; r = 0

8 2 9 5 6 1 2 9 0 3 5 0 7 6 0 6

1 4 5 9 2 5

22

a) 123 · 48 = 5 904 c) 1 482 : 57 = 26

b) 18 · 86 = 1 548 d) 1 862 : 14 = 133

23

a) 6 c) 2 e) 2

b) 3 d) 18 f) 1

24 a) 12

b) 16 d) 28 f ) 40 h) 68

c) 24 e) 34 g) 42 i) 84 25

6 · (8 + 2) = 6 · 8 + 6 · 2 = 60 5 · (9 – 6) = 5 · 9 – 5 · 6 = 15 (10 – 8) · 4 = 10 · 4 – 8 · 4 = 8 (7 – 2) · 12 = 7 · 12 – 2 · 12 = 60 We used the distributive property.

26

a) 100 : 5 = 20 barrels b) 12 · 5 = 60 Euros c) 10 · 24 = 240 bottles d) 360 : 4 = 90 Euros

17

Unit 1

EXERCISES AND PROBLEMS 27

28

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.

32

31

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

8:4+7–3 10 – 12 : 6 – 4 8 + 10 : 5 – 10 11 – 2 – 9 : 3 15 : 3 + 7 + 4 : 2 12 : 4 – 1 – 6 : 3 9:3+8:4–7:7 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

f ) 4 · (7 – 5) + 3 · (9 – 7)

g) 3 · 5 – 3 · (10 – 4 · 2)

h) 2 · 3 + 5 · (13 – 4 · 3)

a) 2; b) 11; c) 47; d) 8; e) 9; f ) 14; g) 9; h) 11

36

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

d) 50 – (16 – 4)

e) 50 + (16 – 4) f ) 50 + 16 – 4

c) 50 – (16 + 4)

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?

35

b

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?

42

Clara was paid €28 to deliver 7 piles of publicity.

How much would she be paid if she delivers an extra pile?

43

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?

44

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?

45

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?

46

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?

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?

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.

34

41

Problem solved

Match each statement to two of the mathematical expressions below:

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 b) d) f) h) j) l) n) p)

e) 2 · (7 + 5) – 3 · (9 – 4) Check your answers:

29

30

b) 5 + 3 · (8 – 6) d) 3 · (2 + 5) – 13

Interpret, describe and express 33

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

Problem solving

Calculate: a) 30 – 4 · (5 + 2) c) 5 · (11 – 3) + 7

— 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?

38

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?

47

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?

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:

39

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?

48

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?

40

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?

49

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?

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?

1. 168 : 2 = 84

2. 84 · 4 = 336

3. 137 · 2 = 274

4. 336 + 274 = 610

5. 714 – 610 = 104

6. 104 : 4 = 26

22

23

Enterprising culture Productivity (productivity dimension): My project.

In exercise 28, students are asked to do some research for which they need to prepare a plan of action in order to be successful.

27

a) True

b) True

c) False

28

D=d·q+r k · D = k · (d · q + r) = k · d · q + k · r The distributive property tells us that the remainder is also multiplied by the same number.

Combined operations 29 a) 20 b) 14 e) 44 f ) 7 30 a) 3

e) 9 i) 3 m) 12

b) 6 f ) 0 j) 14 n) 4

Unit 1

28

31

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

8:4+7–3 10 – 12 : 6 – 4 8 + 10 : 5 – 10 11 – 2 – 9 : 3 15 : 3 + 7 + 4 : 2 12 : 4 – 1 – 6 : 3 9:3+8:4–7:7 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

22

b

b) 5 + 3 · (8 – 6)

c) 5 · (11 – 3) + 7

d) 3 · (2 + 5) – 13

e) 2 · (7 + 5) – 3 · (9 – 4)

f ) 4 · (7 – 5) + 3 · (9 – 7)

g) 3 · 5 – 3 · (10 – 4 · 2)

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

36

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.

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

d) 50 – (16 – 4)

e) 50 + (16 – 4) f ) 50 + 16 – 4

c) 50 – (16 + 4)

Which equation or equations solve the following problem?

42

Clara was paid €28 to deliver 7 piles of publicity.

How much would she be paid if she delivers an extra pile?

43

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?

44

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?

45

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?

46

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?

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?

— 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?

38

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?

47

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?

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:

39

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?

48

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?

1. 168 : 2 = 84

40

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?

49

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?

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?

35

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?

Match each statement to two of the mathematical expressions below:

III. Ernest buys a t-shirt for €16 and a hat for €4.

He pays with a €50 note.

34

41

Problem solved

I. There are 50 people on a bus. 16 people get off the bus at the first stop and 4 people get on.

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

Problem solving

Calculate: a) 30 – 4 · (5 + 2)

Interpret, describe and express 33

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 30

32

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?

2. 84 · 4 = 336

3. 137 · 2 = 274

4. 336 + 274 = 610

5. 714 – 610 = 104

6. 104 : 4 = 26

23

c) 4 g) 3 k) 8 o) 0

d) 4 h) 6 l) 0 p) 2

a) 30 – 4 · 7 = 30 – 28 = 2 b) 5 + 3 · 2 = 5 + 6 = 11 c) 5 · 8 + 7 = 40 + 7 = 47 d) 3 · 7 – 13 = 21 – 13 = 8 e) 2 · 12 – 3 · 5 = 24 – 15 = 9 f ) 4 · 2 + 3 · 2 = 8 + 6 = 14 g) 15 – 3 · (10 – 8) = 15 – 3 · 2 = 15 – 6 = 9 h) 6 + 5 · (13 – 12) = 6 + 5 · 1 = 6 + 5 = 11

b) 9 – (3 + 1) = 5

III → a) and c)

IV → e) and f )

and c)

1 The number of cows is equal to half the number of horns: Cows → 168 : 2 = 84 2nd Cow legs → 84 · 4 = 336 3rd The number of chicken legs is double the number of beaks: Chicken legs → 137 · 2 = 274 4th Cow legs + chicken legs → 336 + 274 = 610 5th The number of horse legs is equal to the total of legs minus cow and chicken legs: Horse legs → 714 – 610 = 104 6th The number of horses is obtained by dividing the previous number by 4: Horses → 104 : 4 = 26 st

Problem solving 36 Problem solved.

37

There are 1 233 valid jars.

38

House B was finished in March.

39

He expects to obtain 4 000 plants.

40 Now 41

there are 118 cans.

She sent 27 messages.

42 If

she delivered an extra pile, she would receive 32 Euros.

43 She

bakes 1 080 muffins a week.

44 There

are 12 horses and 24 cows.

45 There

are a total of 648 cans.

46 The

family earns 6 440 dollars monthly.

47 Julian

18

d) 30 h) 4

32

35

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.

c) 4 g) 6

a) 9 + (3 – 1) = 11

34 b)

EXERCISES AND PROBLEMS

e) False

31

Interpret, describe and express 33 I → b) and d) II → a) and c)

27

d) True

is 12 years old, Rosie is 14 years old and Albert is 16 years old.

48 The

distance between them after half an hour is 80 km.

49 The

distance between them after an hour and a half is 15 km.

Unit 1

EXERCISES AND PROBLEMS 50

51

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?

59

1st

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?

53

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

54

The local tourist industry employs 12 845 people.

Three in every five employees are women. How many female employees are there?

55

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?

57

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?

58

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.

2.ª 1

1

0

111 110 101 100

64

65

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

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 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.

70

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?

71

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?

Draw a grid to help you. For example: 20

30

15

20

20 15

30

20

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

1st 2nd 3rd 4th 62

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?

56

3rd 3.ª 1 0 1 0

2nd

1.ª

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?

52

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

66

A B C D

B D C

C B D

C D B

D B C

D C B

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

Draw a Venn diagram like this one to help you.

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?

GREY

WHITE

GREEN

BLUE

RED

63

Victoria has a farm with ducks and geese. Today, she sold 21 of her animals for €350.

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?

68

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?

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?

YEAR 1 CAFETERIA

OTHERS

How many red cars were produced?

SCHOOL BUS

EXTRA ACT.

72

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?

73

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?

24

25

Page 25 includes a set of problems that use ‘different’ strategies and resources to those intended to contextualise and use the four basic operations. That is, they require a variety of skills in order to solve the problems. Namely: • Critical thinking: 64, 73 • Using grapical resources: 65, 67, 70, 71 • Interpreting graphs: 66 • Generalising processes: 68 • Experimentation, trial and error: 65, 69 • Logic, creativity, imagination: 68, 69, 70, 72, 73 50 The

van still has more than half of its load.

51

They need 14 taxis.

52

It will take her 4 week to buy the skateboard.

53

It makes 174 cars every day.

54 There 55

are 7 707 women.

180 study a second language.

60 study German.

56 He

expects to obtain €9 800 of profit.

57

Marta has spent €66, Julian €36 and Rosie, €106.

58

There are 10 ways of making €1: 10 cts.

0

1

2

3

4

5

6

7

8

9

10

20 cts.

5

0

2

4

1

3

0

2

–

1

–

0

50 cts.

0

2

1

0

1

0

1

0

–

0

–

0

59 There

are 8 four figure numbers that only contain 0 and 1.

There are 16 five figure numbers that only contain 0 and 1.

60 You

51

52

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?

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

The local tourist industry employs 12 845 people.

Three in every five employees are women. How many female employees are there?

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?

57

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?

58

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.

24

1

1

0

111 110 101 100

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

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?

64

65

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?

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.

70

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?

71

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?

Draw a grid to help you. For example: 20

30

15

20

20 15

1st 2nd 3rd 4th 62

66

A B C D

B D C

C B D

C D B

D B C

D C B

30

20

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

Draw a Venn diagram like this one to help you.

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?

GREY

WHITE

GREEN

BLUE

RED

OTHERS

How many red cars were produced?

67

63

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?

68

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?

They can sit in 24 different ways.

62

She needs to order 32 boxes to fill the order.

63

Each duck costs €10, and each goose costs €30. travels 1 950 m in 78 seconds. Yes, the driver exceeded the speed limit.

‘+’ problems

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

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?

56

3rd 3.ª 1 0 1 0

nd 22.ª

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

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?

54

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

1st 1.ª

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?

53

55

59

61

64 It

Unit 1

EXERCISES AND PROBLEMS 50

can choose 30 possible menus.

YEAR 1 CAFETERIA

65 1 624

trees can be planted.

66 3 960

red cars were produced.

67

SCHOOL BUS

EXTRA ACT.

72

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?

73

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

20 % of people surveyed did not go to the beach or stay in the village.

68 The

sum of the numbers one to one hundred is 5 050.

‘+’ problems 69 Open answer.

For example: 1 102, 2 011 or 3 001.

70 It

takes Gemma 10 minutes to catch up with Fred.

71

In year 1 there are 105 students.

72

They weigh 46 kg, 45 kg, 42 kg and 38 kg.

73

The green motorcycle will overtake the red motorcycle in 55 and a half laps.

19

Maths Workshop Unit 1

MATHS WORKSHOP 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 ⎯→

18 →

←• 2 ⎯→

36 →

←• 4 ⎯→

72 →

— 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.

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

←• 16 ⎯→ 288 →

— 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

→ 23

• Use this method to solve the following multiplications:

8

144 414 ←

a) 17 × 41

b) 41 × 17

INVESTIGATE

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

0 1 5

6

2 7

3

• 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

1 4

9

14

triangles

Not possible

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 = 40

a) 154 ·

= 462

c) 30 275 :

b)

· 100 = 27 000

d)

: 10 = 38

b)

= 35

· 125 + 8

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

The history of counting

6 Read the following statements:

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 =

a) 12 + 3 · 5 – 2

b) 131 – 6

3

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

15 16 17 18 19

c) 4 000 :

2

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

4

8

10 11 12 13 14

mayan decimal

4 Solve the following combined operations: +15

squares corners

anayaeducacion.es Answer key.

numeral systems

egyptian

Read and learn

Complete the table below to help you:

remaining corners

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

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.

27

Linguistic Plan Skill: Reading comprehension (expository text).

In this section, students must extract and understand the information contained in the text. Then, they must be able to solve the proposed activities.

Enterprising culture Initiative (productivity dimension): I support changes.

Think about how to solve the activities in this section as a group and reach an agreement, changing opinion if necessary.

Support for comprehensive reading, curiosity about the historical development of maths, through interesting facts or stories related to the evolution of numbers, to calculation strategies and tools, or to men and women who have contributed to the development of maths, all form part of this content which, in addition to being necessary for students’ learning, help us motivate students, break monotonous work routine and make the subject more appealing.

Reading comprehension can be complemented with a round of comments, where students can express their opinions or provide more information about the topic. Lastly, students can be asked to work as a group to find more information (Internet, library…) and prepare a mural showing the development of calculation tools throughout history.

Multiplication in ancient Egypt

You can strengthen the justification of the method by using the following process based on the distributive property: 23 · 18 ↔ (1 + 2 + 4 + 16) · 18 = 1 · 18 + 2 · 18 + 4 · 18 + 16 · 18 = = 18 + 2 · 18 + 2 · 2 · 18 + 2 · 2 · 2 · 2 · 18 • a) 17 × 41 ←• 1 ⎯→ 2 ⎯→

b) 41 × 17 ←• 1 ⎯→

41 →

17 →

82

2

34

4 ⎯→ 164

4

68

8

←• 8 ⎯→ 136 →

328

←• 16 ⎯→ 656 → → 17

16

697 ←

272

←• 32 ⎯→ 544 → → 41 697 ←

Investigate These activities, in which students are presented with a difficulty suitable for their level, but without a theoretical presentation, are usually well received and can be adapted to group work and learning among equals.

To do these activities, students must have an abacus that enables them to experiment and perform trial and error to reach conclusions. Once they discover how the abacus works, it is important to write down these conclusions.

We recommend doing the proposed activities in small groups, without giving prior instruction, followed by a class discussion.

326 326 + 15 341 • 300 + 20 + 6 300 + ( 20 + 10 ) + (6 + 5 ) 300 + ( 30 + 10 ) + 1 300 + 40 + 1

+ 15 211 200 + 10 + 1

• a)

211 + 42

131 – 6 20

253

+ 42 preparation

131 100 + 30 + 1

b)

200 + ( 10 + 40 ) + (1 + 2 )

100 + 20 + ( 11 – 6 )

100 + 20 + 11

–6

–6

125

Unit 1

MATHS WORKSHOP 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 ⎯→ ←• 4 ⎯→

— 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 → 72 →

←• 16 ⎯→ 288 →

— 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

→ 23

• Use this method to solve the following multiplications:

8

144 414 ←

a) 17 × 41

b) 41 × 17

INVESTIGATE

squares

• 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

1

14

triangles

Not possible

anayaeducacion.es Answer key.

0 1 5

6

2 7

3

4

8

9

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 :

b)

· 100 = 27 000

d)

: 10 = 38

b)

= 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)]

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

more chairs than stools. How many chairs are there?

How many stools?

b) 131 – 6

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

3

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

numeral systems

egyptian

2

4

remaining corners

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

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.

Practice makes perfect!

A series of problems or challenges are included in this section that, regardless of theoretical formulations and content programme, aim to make students practise personal preparation strategies when solving logic problems. Of course, students will draw upon their mathematical knowledge, but will also use experimentation, guesswork, trial and error, or any other process that will lead them to the solution.

• By putting 5 in the middle:

a) The honey is placed into half kilo jars. How many jars of honey does she produce each year?

c) Round this number.

6

27

ICT anayaeducacion.es Answers for the self-assessment.

3 1

1 2 3 1 2 3 1 2 3

2

...

1

2

2

5 8

3 4

9 • There are three options for the first figure (1, 2 or 3). For each of these three options, there are another three for the second figure, and another three for the third. Therefore, there are 3 · 3 · 3 = 27 different numbers with the given conditions. Look at the image on the left.

• When trying to complete the number of squares (first row) and the corners that correspond to each case (second row) in the table, the number of triangle corners (third row) is conditioned, which must be a multiple of three. This is the only possible case: 3 squares (12 corners) and two triangles (6 corners). squares

1

2

3

4

5

corners

4

8

12

16

20

remaining corners

14

10

6

2

triangles

Not possible

3 1 3

2

7

SDG commitment Watch the video for target 13.3.

Start a class discussion on actions we can do to slow climate change.

1

1

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?

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.

2 3

2

Therefore, there were four sandwiches and one was cut.

Self-assessment 1

0 1 5

egyptian

2

7

3

4

8

9

10 11 12 13 14

mayan decimal

6

numeral systems

3 042

15 16 17 18 19

13

528

The Egyptian numeral system is additive. The Mayan numeral system is part additive, part positional. And the decimal numeral system is positional.

2

a) 10

b) 270

c) 100

d) 380

3

a) 3

b) 2 646

c) 865

d) 12

b) 32

c) 15

d) 130

4 a) 5

25

There are 15 stools and 45 chairs.

6 a) 8 514 877 → Eight

million, five hundred and fourteen thousand eight hundred seventy-seven b) 8 510 000 → 7 601 770 000 c) Round to the nearest hundreds of thousands: 8 500 000 Round to the nearest hundreds of millions: 7 600 000 000

7

There is a distance of 25 km between them after 10 minutes.

8

a) She produces 6 732 jars of honey per year. b) She makes an annual profit of €20 196. c) The rounded profit is €20 000.

21